Objectives. 3.3 Toward statistical inference

Transcription

1 Objectives 3.3 Toward statistical inference Poulation versus samle (CIS, Chater 6) Toward statistical inference Samling variability Further reading: htt://onlinestatbook.com/2/estimation/characteristics.html (some of the concets introduced in this link are beyond this class) Adated from authors slides 2012 W.H. Freeman and Comany

2 Amazon reviews

3 Which tracker should I buy. How to comare them? The smart Lintelek has the best average rating of 4.8 stars. But the samle size for the smart Lintelek is on 58. The other two trackers, have worse reviews (average 4.2 and 4.4), but their samle sizes are substantially larger, 174 and 261. How can we comare these numbers? If more eole are asked, the averages will change. Ideally, we could comare the true averages (mean based on the oulation). This way we can definitively say that eole refer one roduct to another based on their mean.

4 Refresher: Definitions Poulation: The entire grou of individuals in which we are interested but cannot assess or observe directly. Samle: The art of the oulation we actually examine and for which we do have data. Examles: consumers who have urchased the roduct. Examles: All online consumers. A arameter is a number describing a characteristic of the oulation. Usually it is the mean, such as the mean rating of a roduct. Poulation Samle A statistic is a number describing a characteristic of a samle. Usually it is the samle average such as an amazon rating.

5 Samle 1: Samling variability The histogram is a lot of the roortion of ratings given for a articular tracking watch. The mean score is 4.2 Here we samle 10 eole and ask them to rate the watch. We see that for this samle all 10 individuals rate the watch 5. Giving a mean of 5! Which is not the same as 4.2.

6 Samle 2: Samling variability The histogram is a lot of the roortion of ratings given for a articular tracking watch. The mean score is 4.2 Here we samle 10 eole and ask all of then to rate the watch. In this samle, 2 eole give a rating of 1, which dros the average to 3.9.

7 Samling variability As illustrated in the revious examle, for every samle taken from a oulation, we are likely to get a different set of individuals and calculate a different value for our statistic (such as the samle mean). This is called samling variability. This would suggest that the samle and the statistic contains no information about the oulation. However. The good news is that, if we imagine taking lots of random samles of the same size from a given oulation, the variation from samle to samle the samling distribution will follow a redictable attern. All of statistical inference is based on this; to see how trustworthy a statistic is what haens of we ket reeating the samling many times?

8 We measure the quality of a statistic (such as the samle mean) with: Accuracy (bias) Random samles rovide accurate estimates of a arameter because they are unbiased (or close to unbiased, deending on the random samling method). This is done by samling in a good way (ie. Randomly samling over the oulation of interest). Tyically we will assume an estimator is unbiased. When reading an article identify the oulation of interest and otentially biases which may arise. Reliability (variable) A reliable estimation method is one that would give similar results if the random samling is reeated over. The less variable a statistic, the more reliable it is. Random samling enables us to measure the variability of a statistic. We do this with the standard error in the next slide we define what this means. Imortant: The larger the samle size, the less variable the corresonding statistic will be. To understand the above concets look at the question at the end of this age: htt://onlinestatbook.com/2/estimation/characteristics.html

9 Measuring Variability We have come across variability before. Recall in Chater 3 we used the standard deviation to measure the variability in the samle. We recall that the samle standard deviation is the deviation from each observation to the samle mean: s = v u t 1 n 1 nx (X i X) 2 i=1 q The same criterion is used to measure the variability in the samle mean (and all other estimators). This is called the standard error. q More recisely, we measure the average sread from each estimator to the true mean. q This sounds imossible. q Remarkably, we can find a very nice exression for the standard error which requires very little effort!

10 What do you mean?? Variability of samle mean is a very strange notion. But let us do a thought exeriment with the aid of Statcrunch. We ask 10 eole to review the Lintelek tracker. The average score amongst these 10 eole eole is 5. We ask another 10 eole to review the Lintelek tracker. The average score amongst these 10 eole eole is 3.9. We ask another 10 eole to review the Lintelek tracker. The average score amongst these 10 eole eole is 4.3. Each of these averages are different, they vary (just like an individual score varies from erson to erson), but because it is an average the variation is less than an individual score. The variation amongst these three averages is r ((5 4.4)2 +( ) 2 +( ) 2 )=0.56 In reality we want to calculate the standard deviation of all ossible averages, this is called the standard error.

11 Poulation size does not matter Question: Would the estimate of the roortion be better if the oulation size were smaller? For examle, 1.5 million students rather than 15 million student. Answer: No. Only the size of the samle, in this case n=2000, has an influence on it s reliability, not the size of the oulation. Statistical inference is not based on how close the samle size is to the oulation (usually we assume that the oulation is infinite). It is based on the idea that simle random samle gives a reresentative samle over the entire oulation.

12 Summary and what s to come The techniques of statistics allow us to draw inference or conclusions about a oulation using the data from a samle. Your estimate of the oulation arameter is only as good as your samling design. à Work hard to eliminate biases (design your exeriment well). Your samle statistic is only an estimate and if you randomly samled again you would robably get a somewhat different result (more of this next). In the next section we will show: q q The distribution of the estimates (for much of the course it will be the samle mean) will (if the samle size is large enough) be normally distributed even if the observations are not normal. The standard error (reliability) has a simle formula!

13 Objectives 5.1 Samling distribution of a samle mean (CIS, Chater 8) The mean and standard deviation of For normally distributed oulations x The central limit theorem (CIS, Chater 8 and 103) Additional reading: htt://onlinestatbook.com/2/samling_distributions/sam_dist_mean.ht ml Adated from authors slides 2012 W.H. Freeman and Comany

14 Toic: Behavior of samle mean Learning objectives: Understand how the samling a in Statcrunch works and why we use it. Understand what drawing a samle from a distribution means, and what samle size means. Understand that the distribution (density curve) of the data can have many different shaes. Samle size cannot change the shae. However, if you take the average of this samle and were able to draw several samles and take the average of each, then the distribution of the average: Will have less sread (variability) then the distribution of the data. Will be more symmetric than the distribution of the data. Will, for large samle sizes, be close to normal (how large deends on how close the original data is to normal). The sread of the samle mean is called the standard error and can be calculated with the formula σ/ n. You will need to use this a later, to check the reliability of the data analysis.

15 Simulation tools used To demonstrate the concets, I will be using an Alet in Statcrunch called samling distribution. It is highly recommended that you try this out yourself. Alets -> Samling Distributions. Select the distribution (from uniform etc) or choose the data table (your own data). Press comute. Choose your samle size (this is how large a samle you use). You will see a 1000 button, this has NOTHING to do with samle size. It is the number of samles you draw. I usually set it to 60K. Press the + sign next to Samling means to get the QQlot of the distribution of the samle mean. Do not ress the + sign next to Samles this will give you the QQlot of the samle. The ideas are rather sohisticated and it will take time to understand them. Note that you can customize the (arent) distribution from which you samle from by simly left clicking over the arent distribution and moving the cursor as you want the shae of the distribution to be.

16 Proortions As a thought exeriment we treat the oulation scores for the rating of the tracker as known. In reality this is never known. Suose 70% of the oulation would rate is 5 stars. 10% 4 stars 3% 3 stars 6% 2 stars 11% 1 star. In the following examles eole from this distribution will be drawn. This will form the samles.

17 Distribution of average: samle 5 Let us now look at the distribution of the samle mean of all samles of size 5. That is we samle 5 eole the oulation ask their ratings and evaluate the samle mean. The true roortions A samle of 5 eole who were questioned and their scores. The average rating of these 5 scores is 4. We do the same thing many, many (36K) times. Every average score out of 5 eole is lotted in this histogram. We have made a sread sheet of 36K averages.

18 QQlot of average: samle 5 Because we have a whole load of averages (we made 36K of them!) we can make a QQlot of these samle means, of all samles of size 5 (corresonding to the histogram on the revious age). Observations: The histogram of the samle mean is more bell-shaed than the original distribution. However, it is certainly not normal. But there is less sread in the distribution of the averages than the original histogram. The QQlot shows a large deviation from normality in the tails.

19 Distribution of average: samle 20 Let us now look at the distribution of the samle mean of all samles of size 20. That is we samle 20 eole from the oulation ask their rating, and take the samle mean. The true roortions A samle of 20 eole who were questioned and their scores. The average rating of these 20 scores is We do the same thing many, many (60K) times. Every average score out of 20 eole is lotted in this histogram.

20 QQlot of average: samle 20 Let us now look at the QQlot of the samle mean of all samles of size 20 (corresonding to the histogram on the revious age) Observations: 1. The histogram of the samle mean is a lot more bell-shaed than the original distribution. The sikes that were seen for samle size 5 have gone (the bums you see on the histogram are due to binwidth choice). 1. There is even less sread in the distribution of the averages than the original histogram. 2. The QQlot shows a deviation from normality in the to tail of the distribution.

21 1. The QQlot shows a some deviation from normality in the to tail of the distribution. 2. This deviation is because the samle mean cannot be greater than 5 (since that is the maximum score), whereas observations from a normal distribution can be.

22 Distribution of average: samle 50 Let us now look at the distribution of the samle mean of all samles of size 50. That is we samle 50 eole from the oulation ask their rating, and take the samle mean. The true roortions A samle of 50 eole who were questioned and their scores. The average rating of these 50 scores is We do the same thing many, many (60K) times. All the average scores are lotted in this histogram.

23 QQlot of average: samle 50 Let us now look at the QQlot of the samle mean of all samles of size 50 (corresonding to the histogram on the revious age) Observations: 1. The histogram of the samle mean is retty much normal. 2. There is even less sread in the distribution of the averages than the original histogram. 3. The QQlot shows only a very tiny deviation from normality in the tails of the distribution. This small deviation is because of the skewness in the original data.

24 Summary: Different samle sizes

25 q Take home message Proortions for the ratings is definitely not normal. q The ratings that each individual gives is numerical discrete (1,2,3,4 or 5). q The histogram shows that it is left skewed, with a large roortion at 5 stars. Standard deviation is σ= q q q If 5 eole are interviewed and the average evaluated, the distribution of the average rating is more symmetric, with less sread 0.614=1.336/ 5. We call the standard error for the samle mean of size 5. If 50 eole are interviewed and the average evaluated, the distribution of the average rating is close to normal with far less sread 0.19=1.336/ 50. We call 0.19 the standard error for the samle mean of size 50. The sread of the samle means decreases as the samle size increases.

26 Formula for standard error If the standard deviation of the original data is The formula for the standard error of the samle mean (standard deviation of samle mean) when the samle size is n is n

27 q How does this hel when choosing a tracker watch? The sread between the samle means decrease as the number in the samle increases. The imlication is that we are more and more likely to close in to the mean as the samle size grows. How to quantify more likely. Usually this would be imossible. But for large samle sizes, the samle mean is close to a normal distribution, centered about the true mean roortion (4.2 in our running examle). In chater 4 we learnt if a variable is normally distributed, then the roortion of the oulation that is within 1.96 standard deviations of the mean is 95%.

28 Alying this result: Near normality of the samle mean imlies that 95% of of the samle means will be within 1.96 standard errors of the oulation mean. The standard error is 1.336/ 50 = If the oulation mean rating of a roduct is 4.2, and 50 consumers are asked to rate the roduct the roortion of average ratings (amongst 50 consumers) that will be within 1.96x1.336/ 50 = 1.96x0.19 = 0.37 of 4.2 is 95%. In reality we do not know the mean rating of a roduct. But now that we understand the roerties of the samle mean, we can use these results to locate the oulation mean (we do this in Chater 6).

29 Proerties: Samle mean for normally distributed data When a variable in a oulation is normally distributed, the samling distribution of distributed. x for all ossible samles of size n is also normally If the oulation is Normal(µ, s) then the samle mean s Samling distribution distribution is Normal(µ, s/ n). Note that the samle average has less variability than any Poulation individual observation.

30 Averages: skewed distributions The salaries of NBL layers is numerical continuous, over 60% earn less than 5million dollars, but a small ercentage earn more than 20million. It is clear that this data is not normal (the QQlot is very S shaed). On the next slide we look at the distribution of the samle means of NBL salaries based on different samle sizes.

31 Proerties: Samle mean of non-normal distributed data Central Limit Theorem: When randomly samling from any oulation with mean μ and standard deviation σ, if n is large enough then the samling distribution of is aroximately normal: ~ N(μ, σ / n). x Samling distribution of x for n = 50 observations Samling distribution of x for n = 500 observations

32 The sread of the samle decreases as the samle size increases. It decreases according to the formula: σ/ n (this is called the standard error). The more skewed the original data, the larger the samle size required for the samle mean to be close to normal.

33 Question Time The distribution of heights has standard deviation 3. A samle of one erson is drawn. What is the standard error for the average based on one? (A) 3/1=3 (B) 3/2=1.5 (C) It is unknown. htt:// A samle of 5 eole is taken. What is the sread (standard error) of the average of 5 heights? (A) 3 (B) 3/ 5 = 1.34 (C) 3/5 = 0.6 htt://

34 Calculations

35 Toic: Calculations Learning targets: Be able to calculate robabilities for averages based on the average being close to normally distributed. Be able to assess is the samle average is close to normal based on the histogram of the data and the samle size. Be able to assess if the robabilities are accurate/correct based on how close the average is to normal. Be able to do sum tye calculations.

36 Normal data: Calculation Practice 1 In 2010 the combined SAT scores had mean 1016 and standard deviation 212. They also had aroximately normal distribution. Poulation distribution is Normal(μ = 1016; σ = 212). In Chater 4, we used the normal distribution to show that the robability of a randomly selected student scoring 1100 or higher is 34.5%. Now, suose 50 students are randomly selected and their SAT scores averaged. What is the robability that the average is greater than 1100? Samling distribution of the samle average when n = 50 is Normal(μ = 1016; σ / n = 212 / 50 = 29.98). Using these values, the z-score for 1100 is z ( x -µ) = = ` = = s n In Table A, the area to the right of 2.80 is So there is only a 0.25% chance that the average of 50 randomly samled students is more than In this examle we do not use the CLT because the original data is assumed normal.

37 On the left is the distribution of the SAT score for one erson. The mean grade is And we see that there is a lot of variation. So 34.6% of students score more than 1100 in their SATs. On the left is the distribution for the average SAT grade over 50 students. This the average grade amongst a class of 50. The x-axis of both lots have been aligned. The average grade over 50 is clustered closer to the global average. There is less variation. The roortion of classes with an average grade over 1100 is very small; only 0.25%. I urosely aligned the x-axis in both lots!

38 Normal data: Calculation Practice 2 Hyokalemia is diagnosed when mean blood otassium levels are below 3.5mEq/dl. Suose a diagnoses is made based on the blood samle. If the blood samle gives a otassium level less than 3.5 a diagnoses is made (later we exlain this is a wrong strategy and gives rise to too many false ositibes). Let s assume that we know a atient whose measured otassium levels vary daily according to the Normal(μ = 3.8, σ = 0.2) distribution (this erson by definition does not have low otassium since μ = 3.8 > 3.5). If only one measurement is made, what is the robability that this atient will be misdiagnosed with Hyokalemia? z ( x -µ) = = =-1.5 s 0.2 P(z < 1.5) = %.

39 Normal data: Calculation Practice 2 Instead, suose measurements are taken on 4 searate days, and the average evaluated. If the average is less than 3.5, low otassium is diagnosed. For the same atient, is the robability of a misdiagnosis? z ( x -µ) = = =-3 s n P(z < 3) = %.

40 Question Time Female heights are normally distributed with mean 65 inches and standard deviation 2 inches. The average of 9 (randomly) samled females is taken. What is the chance that their average height will be less than 63 inches? (A) 99.87% (B) 0.13% (C) 3% (D) 15.8% htt://

41 Non-normal data: Calculation Practice In Chater 4 we discussed ACT scores. We argued that because the grades were numerical discrete over a small range the grade distribution could not be normally distributed. BUT if the samle size is large enough the average will be close to normal. We recall the mean ACT score is 22 with standard deviation 5. Question: 50 students are randomly selected and the average taken. Calculate the roortion of averages which are greater than 20. Answer: The mean of the samle mean has the same mean as the original distribution, which we know is 22. The standard error of the samle mean is 5/ 50 = On the left we have the histogram of ACT scores on the right we have the histogram of the average based on 50 uils. The histogram of the average looks more normal.

42 Answer: The mean of the samle mean has the same mean as the original distribution, which we know is 22. The standard error of the samle mean is 5/ 50 = We use this to make the z-transform z = = 2.82 Looking u the z-tables using a comuter we see that robability is 99.7%. This means there is a very large chance the samle mean of a class of size 50 is greater is than 20.

43 Nonnormal data: Calculation Practice 2 Let us return to the weights of calves at 0.5 weeks. q Looking at the lot, it seems that a normal density (with mean and standard deviation 7.7) is a rough aroximation of the underlying distribution of calves weights (see also the QQlot given at the end of Chater 4). q q Question 1: Using the normal density calculate the roortion of calves that weight more than 100 ounds. Answer: Make a z-transform=( )/7.7 =1.28. This corresonds to 90% in the z-tables. Therefore, if the calf weights follow a normal distribution 10% of 0.5 week year calves will weight more than 100 ounds.

44 Question (b): Let us suose that the samle mean of 10 calves is taken. From the histogram of the samle mean, we see that it is close to normal. Using the normal distribution, what roortion of the samle means (based on samles of size 10) will be greater than 100 ounds? Answer: The mean of the samle mean is the same as the mean weight of cows which is The standard error of the samle mean is 7.7/ 10 = 2.4. By making the z-transform we have z=( )/2.4 = Looking u the z-tables, we see that it is in the far uer tails; the robability is close to 0%.

45 Concetual understanding q Of the two calf robabilities calculated above, which is likely to be closest to the roortion calculated using the correct distribution? q q Both robabilities were calculated using the normal distribution. But this is only an aroximation of the true distribution of calf weights and samle mean of calf weights. From the histogram of calf weights we see that is only aroximately normal. This means it is unlikely that the roortion calculated for the weight of one calf is that recise. On the other hand the Central Limit Theorem tells is that the distribution of the samle mean gets closer to normal as the samle size grows. The second roortion we calculated was based on the average weight of 10 calves. The distribution of the average is closer to normal than the distribution of one calf. Thus the second roortion will be closer to the roortion corresonding the distribution of the average.

46 Question Time (windchill 1) Wind chill is the erceived decrease in air temerature felt by the body due to wind. The mean is -28 and the standard deviation is 36. Assuming normality of the average, calculate the chance the average wind chill factor over 3 consecutive days will be more than zero. htt:// A. 77.7% B. 78.1% C. 8.9% D. 91.1%

47 Question Time (windchill 2) Base on the lot and QQlot of the data, is the robability calculated on the revious slide close to the true robability calculated if one were able to obtain the histogram of all averages of samle size 3? A. Yes, because the average will be normal. B. No, because the data is (thick tailed), clearly not normal. So an average based on 3 is not enough to invoke the CLT. htt://

48 Windchill average based on 3 We draw many samles each of size 3 from the windchill data and evaluate the average for each samle. A QQlot of these averages is given below. We see that the average based on three is still not normally, and there is a large deviation in the tails, esecially larger than 0 (for the samle mean). This means the robability calculated two slides back, which is based on the normal distribution will not be close to the true robability (calculated from all averages).

49 Windchill average based on 3 The lack of normality means the robability calculated two slides back, which is based on the normal distribution, will not be close to the true robability (calculated from all averages). The to gives the true histogram of the averages the bottom gives the normal aroximation. We see the shaes do not match.

50 Windchill average based on 30 We draw many samles each of size 30 from the windchill data and evaluate the average for each samle. A QQlot of these averages is given below. We see that the average based on 30 is very close to normal. This means the roblems which involve the average over 30 days and robabilities calculated using the normal distribution are likely to accurate.

51 Alication to sum roblems

52 Sums: Calculation ractice A farmer wants to use a vehicle to carry week old calves. The vehicle he lans to use can carry a maximum load of 2760 ounds. He knows that the mean weight of a calf is ounds and the standard deviation is 7.7. What is the chance the vehicle can carry the calves? We need to transform the total weight into an average (samle mean). We observe, if the total weight of 30 calves needs to be less than 2760 ounds this is the same as the samle mean weight of 30 calves must be less than 2760/30 = 92: X30 i=1 X i < 2760 ) X = 1 30 X30 i=1 X i < Therefore, we have turned the roblem from totals into averages and aly the CLT to calculate the robability using the normal distribution.

53 Calculation ractice (cont) We know from the central limit theorem that the samle mean is close to normally distributed. Thus the distribution of the samle mean is normal with mean and standard deviation 7.7/ 30 = 1.4. We know that for the vehicle to carry the calves, the samle mean has to be less than 92 ounds. Calculate the z-transform z=( )/1.4 = 1.35 and look u the z- tables to get Conclusion: 91.1% of the time the vehicle the will be carrying 30 calves legally. P X30 i=1 X i < 2760! = P X = 1 30 X30 i=1! X i < = P (Z <1.35) = 0.911

54 Question Time Suose the mean number of cans of lemonade consumed er erson at a arty is 3 and the standard deviation is one. A host is lanning a arty and is trying decide how many cans of lemonade to urchase for the arty. Suose 200 eole attend the arty. Using the normal distribution (since averages based on a samle size of 200 may be close to normal), what is the chance/robability that they will need to urchase more than 700 cans of lemonade? (A) 0% (B) 0.9% (C) 99.1% (D) 100% htt://

55 How large is a large enough samle size? It deends on the oulation distribution. More observations are required if the oulation distribution has a large standard deviation or if it is far from normal in distribution. A samle size of 25 is generally enough to obtain a normal samling distribution from a oulation with some skewness or even mild outliers. A samle size of 40 will tyically be good enough to overcome some skewness and outliers. More imortantly, n should be large enough to make the standard error sufficiently small then we can get meaningful and recise inferences. We can check this by using the Samling distribution alet. In many cases, even n = 40 is not large enough to give results reliable enough when there is a lot at stake. This is why clinical trials, olitical olls and marketing surveys tyically observe 100 s or even 1000 s of individuals.

56 The effect of skewness on the CLT Below we look at the samle mean taken from data with a large right skew This distribution is clearly right skewed. This is is the distribution of the average of 20 observations drawn from the above right skewed distribution. We see that the means of both distributions are aligned (as we would exect). The distribution of the average is less skewed, but it still is skewed.

57 The corresonding QQlot of the samle mean Observations: 1. The QQlot deviates from normality in the tables, esecially in the tails. The distribution of the samle mean still has a slight right skew (look back at the QQlots in Chater 4). This demonstrates that when data is highly skewed, we need a much large samle size for the CLT to kick in. 2. Calculations based on normality of the the average will not be comletely correct.

58 Effect of binary data on the CLT Binary data arises in several situations where ever there are only two ossible choices eg. Like or Dislike. In this examle, we have encryted one outcome with zero and the other with 1 (it does not really matter which way). We see that the roortion in the one category is about 20% - this is what is meant by the mean. This data is discrete and clearly skewed.

59 The corresonding QQlot of the samle mean Observations: 1. We see that the standard error is = 0.405/ 50, which is as it should be. 2. However, the QQlot deviates far from normality in the tables. The lines across demonstrate that the average over 50 still takes discrete values (though not integers). We also see a U shae that shows that the samle mean is still skewed. 3. Calculations based on normality of the the average will not be comletely correct.

60 Question Time Let s consider the very large database of individual incomes from the Bureau of Labor Statistics as our oulation. Income is strongly right skewed. Which histogram corresonds to samles of size We take 1000 SRSs of 25 incomes, calculate the samle mean for each, and make a histogram of these 1000 means. We also take 1000 SRSs of 100 incomes, calculate the samle mean for each, and make a histogram of these 1000 means. 100? Which to samles of size 25? (A) Left = samle size 25, Right = samle size 100 (B) Left = samle size 100. Right = samle size 25. htt://

61 So many standard deviations! In statistics we talk about different kinds of standard deviations, and it can be hard to kee track of them: s is the standard deviation of a set (samle) of data. It is a statistic we can comute once we have the data. σ is the standard deviation of a oulation (which is much too big to observe comletely). It is a arameter usually, we will never know its true value. σ /Ön is the standard deviation of the values of from all ossible random samles of size n. It refers to the samle mean, not to data. It is also called the standard error of. s /Ön is our estimate of σ /Ön, since we do not know the value of σ. From a survey of students taking statistics, n = 459 resonded to the question How many Facebook friends do you have? The samle mean was x = and the samle standard deviation was s = The standard error for the samle mean is s /Ön = 589.5/Ö459 = x is an estimate for μ = mean of the oulation of all students required to take the class and s is an estimate for the oulation standard deviation σ. x x

62 Summary is always unbiased for μ, even if the oulation s distribution is very different from a normal distribution. The standard deviation of, σ / n, measures the variability due to random samling. If the oulation is aroximately normal or if the samle size n is large, we can use the normal distribution to comute robabilities for. We just have to remember to use σ / n, not σ, in the denominator when calculating z. This means we can say something about how close is likely to be x x x to μ. Generally it is quite likely (95% chance) that it will be within 2 standard errors of μ. Not all variables are normally distributed and large samles are not always attainable. In such circumstances, a statistician should be consulted for roer methods of statistical inference and calculation. x

63 Accomanying roblems associated with this Chater Quiz 5 Quiz 6 Homework 2, Q6. Homework 3.