Chapter 6 Confidence Intervals Section 6-1 Confidence Intervals for the Mean (Large Samples) Estimating Population Parameters VOCABULARY: Point Estimate a value for a parameter. The most point estimate of a population parameter is the sample. Interval Estimate - An, or of, used to estimate a population. Level of Confidence - Denoted as, it is the that the interval estimate the. Margin of Error - Sometimes also called the error of estimate, or error. It is denoted as, and is the possible between the estimate and the value of the it is estimating. c-confidence interval - Is found by and from the sample. The that the confidence interval contains is. FORMULAS: Margin of Error σ E z c σ x z c z c n To use the formula, it is assumed that the standard deviation is known. This is the case, but when n 30, the standard deviation can be used in place of. s The formula effectively becomes E z c of 1 c or 1 n (1 c) (explained why on bottom of page 311) c-confidence interval - sample standard deviation: x E < μ < x + E s (x x ) ; It is MUCH easier to use the STAT Edit function of the calculator to find s. n 1 To find the confidence interval on the calculator: We are going to walk through Example 4 on page 314, using the data points from Example 1 on page 310. 1) Enter the 50 data points into L1 on your calculator (STAT Edit). ) STAT Calc 1 to find the sample standard deviation (we can use this because we have 50 data points; n 30). s 3) STAT TESTS 7 (Z-Interval) 4) Select Data, since you have the data entered into the calculator. 5) Enter as the standard deviation, and as the C-Level (level of confidence). 6) Select Calculate to get the interval. We can be 99% sure that the actual population mean is between and. Notice that the calculator also tells us that the mean of the data we entered is 1.4, that the standard deviation of the data was 5.01 and that n was 50. If we need to know what E is, simply find the distance between the interval endpoints and divide by. (14.5 10.575)/. We could also simply find the distance between the sample mean (if we know what it is) and the endpoint(s) of the interval to find E. 14.5 1.4 1.85 1.4 10.575 1.85
Look at Example 5 on page 315. n 0, x.9, σ 1.5, and c 90% STAT TESTS 7, select Stats (since you are going to provide the stats instead of the actual data points). Enter 1.5,.9, 0, and.9 and then calculate. We can be 90% certain that the actual mean age of all students currently enrolled in college is between and. As a general rule, we round our interval endpoints to the same number of decimal points as the mean that is given to us. We were given.9, which is one decimal, so we rounded our interval to one decimal place. CALCULATING MINIMUM SAMPLE SIZE How do you know how many experiments or trials are needed in order to achieve the desired level of confidence for a given margin of error? We take the formula for finding E and solve it for n. s E z c becomes n z n c( σ E ). Remember, if you don t know what σ is, you can use s, as long as n 30. Example 6 on page 316- c 95%, z c 1.96, σ s 5.01 (from Example 1), E 1 (given). n ( z cσ E ) ( 1.96(5.0) ) ( 9.8 1 1 ) 9.8. If you want to be 95% certain that the true population mean lies within the interval created with an E of 1, you need AT LEAST magazine advertisements in your sample. We round up, since advertisements are not quite enough. Section 6- Confidence Intervals for the Mean (Small Samples) Estimating Population Parameters The t-distribution: What do we do if we don t know the population standard deviation, and can t find a sample size of 30 or more? If the random variable is normally distributed (or approximately normally distributed), you can use a t- distribution. t-distribution formula: t x μ s n critical values of t are denoted as t c just as critical z values are called z c. Properties of the t-distribution (Page 35) 1) The t-distribution is and about the. ) of are equal to n-1. 3) The total area under the curve is, or. 4) The,, and of the t-distribution are equal to. 5) As the degrees of freedom, the t-distribution approaches the distribution. After d.f., the t-distribution is very close to the standard normal z-distribution. Close enough, in fact, that we use the standard normal distribution for d.f. 30. We just did that in Section 6-1.
The TI-84 can also do a t-distribution interval for you!! STAT TESTS 8 (T-Interval) Same as with the z-interval, if you have the data points, select data. If you have the statistics, select stats. Enter the appropriate numbers and select Calculate. EXAMPLE 1 (Page 36) Find the critical value t c for a 95% confidence level when the sample size is 15. If n 15, then the degrees of freedom are If c.95, the area is 1 (1.95) (this is the same formula we used to find the area for z c in Section 6-1. nd VARS 4, 0.05, 14, Enter gives us a t c of (use the positive of the number you get). EXAMPLE (Page 37) You randomly select 16 coffee shops and measure the temperature of coffee sold at each. The sample mean temperature is 16.0 0 F with a sample standard deviation of 10.0 0 F. Find the 95% confidence interval for the mean temperature. Assume the temperatures are approximately normally distributed. We MUST use the for this; the sample size is than and we don t know what is, but we do know that the distribution is. STAT TESTS 8, select Stats, and enter the values for x, s, n, and c and calculate. The 95% confidence interval is from to. We round to decimal because we were given x rounded to decimal. SO, we are 95% confident that the actual population mean of coffee temperature in ALL coffee shops is between 0 F and 0 F. EXAMPLE 3 (Page 38) You randomly select 0 mortgage institutions and determine the current mortgage interest rate at each. The sample mean rate is 6.%, with a sample standard deviation of 0.4%. Find the 99% confidence interval for the population mean mortgage interest rate. Assume the interest rates are approximately normally distributed. We MUST use the for this; the sample size is than and we don t know what is, but we do know that the distribution is. STAT TESTS 8, select Stats, and enter the values for x, s, n, and c and calculate. The 99% confidence interval is from % to %. We round to decimals because we were given x rounded to decimals. SO, we are % confident that the actual population mean mortgage interest rate for ALL mortgage institutions is between % and %. Now, find the 90% and 95% confidence intervals for the population mean mortgage interest rate. What happens to the widths of the intervals as the confidence levels change? There is a flow chart on how to decide which distribution to use (t or normal) on page 39. Study this!! EXAMPLE 4 (Page 39) You randomly select 5 newly constructed houses. The sample mean construction cost is $181,000 and the population standard deviation is $8,000. Assuming construction costs are normally distributed, should you use the normal distribution, the t-distribution, or neither to construct a 95% confidence interval for the population mean construction costs? Explain your reasoning. Although, we can still use the distribution because we know what is. You randomly select 18 adult male athletes and measure the resting heart rate of each. The sample mean heart rate is 64 beats per minute with a sample standard deviation of.5 beats per minute. Assuming the heart rates are normally distributed, should you use the normal distribution, the t-distribution, or neither to construct a 90% confidence interval for the mean heart rate? Explain your reasoning.
Because, the distribution is, and we do know what is, we should use the -distribution on this one. Section 6-3 Confidence Intervals for Population Proportions Sometimes we are dealing with probabilities of success in a single trial (Section 4-). This is called a. In this section, you will learn how to a population proportion p using a confidence interval. As with confidence intervals for µ, you will start with a. The point estimate for p, the population proportion of successes, is given by the proportion of successes in a sample and is denoted by p x, where x is the number of in the sample and n is the number in the. n The point estimate for the proportion of is q 1 p. The symbols p and q are read as p hat and q hat A c-confidence interval for the population proportion p is p E < p < p + E, where E z c p q n. The probability that the confidence interval contains p is c. In Section 5-5, you learned that a binomial distribution can be approximated by the normal distribution if np 5, and nq 5. When np 5 and nq 5, the sampling distribution is approximately normal with a mean of μ p p and a standard deviation of σ p pq n. Guidelines (Page 335) Constructing a Confidence Interval for a Population Proportion. 1) Identify the sample statistics n and x. ) Find the point estimate p. 3) Verify that the sampling distribution of p can be approximated by the normal distribution. 4) Find the critical value z c that corresponds to the given level of confidence c. 5) Find the margin of error, E. 6) Find the left and right endpoints and form the confidence interval. As with the other intervals we ve discussed, the calculator will also create a proportion interval for you. STAT - TESTS A (1-PropZInt) Enter x, n, and the confidence level get the interval. Finding a Minimum Sample Size to Estimate p. Given a c-confidence level and a margin of error E, the minimum sample size n needed to estimate p is: n p q ( z c E ). This formula assumes that you have a preliminary estimate for p and q. If not, use 0.5 for both. EXAMPLE 1 (Page 334) In a survey of 119 U.S. adults, 354 said that their favorite sport to watch is football. Find a point estimate for the population proportion of U.S. adults who say their favorite sport to watch is football. If n and x, then p x n, or % In a survey of 1006 adults from the U.S., 181 said that Abraham Lincoln was the greatest president. Find a point estimate for the population proportion of adults who say that Abraham Lincoln was the greatest president. If n and x, then p x n, or %
EXAMPLE (Page336) Construct a 95% confidence interval for the proportion of adults in the United States who say that their favorite sport to watch is football. First, check to be certain that np and nq. ( )( ) and ( )( ). Find the margin of error; E z c p q, E 1.96 (.9)(.71). n 119 Now that we know that the margin of error is, we that from to get the end of the interval, and it to to get the end of the interval. - < p < + ; < p <. EXAMPLE 3 (Page 337) According to a survey of 900 U.S. adults, 63% said that teenagers are the most dangerous drivers, 33% said that people over 75 are the most dangerous drivers, and 4% said that they had no opinion on the matter. Construct a 99% confidence interval for the proportion of adults who think that teenagers are the most dangerous drivers. To find x, you need to multiply the ( ) by the ( ) to get. p x which means that x p n n n is given to us, at. p is also given to us, at. This makes q, or. We memorized the critical z score for a 99% confidence interval ( ). Putting all of this together, E z c p q, E.575 (.63)(.37) n 900 Now that we know that the margin of error is, we that from to get the end of the interval, and it to to get the end of the interval. - < p < + ; < p <. EXAMPLE 4 (Page 338) You are running a political campaign and wish to estimate, with 95% confidence, the proportion of registered voters who will vote for your candidate. Your estimate must be accurate within 3% of the true population. Find the minimum sample size needed if (1) no preliminary estimate is available and () a preliminary estimate gives p 0.31. Compare your results. n p q ( z c E ). Remember to use.5 for both p and q when you have no preliminary data. 1) n (. 5)(.5)( 1.96 0.03 ). ) n (. 31)(.69)( 1.96 0.03 ) You need a sample size if you don t have any data. Section 6-4 Confidence Intervals for Variance and Standard Deviation In manufacturing, it is necessary to control the amount that a process varies. For instance, an automobile part manufacturer must produce thousands of parts to be used in the manufacturing process. It is important that the parts vary little or not at all. How can you measure, and thus control, the amount of variation in the parts? You can start with a point estimate. The point estimate for is and the point estimate for is. is the most unbiased estimate for. You can use a to construct a confidence interval for the and deviation.
If the random variable x has a distribution, then the distribution of X (n 1)s σ as long as n >. There are 4 properties of the chi-square distribution: 1) All chi-square values X are greater than or equal to. ) The chi-square distribution is a family of curves, each determined by the of To form a confidence interval for σ, use the X -distribution with degrees of freedom equal to the sample size. d.f.. 3) The area under each curve of the chi-square distribution equals. 4) Chi-square distributions are. less than There are critical values for each level of confidence. The value X R represents the critical value and X L represents the critical value. Area to the right of X R 1 c and the area to the right of X L 1+c Table 6 in Appendix B lists critical values of X for various degrees of freedom and areas. Each area in the table represents the region under the chi-square curve to the of the critical value. Confidence Intervals for σ and σ. You can use the critical values X R and X L to construct confidence intervals for a population variance and standard deviation. The formula for the confidence interval for a population variance σ is: (n 1)s X < σ < (n 1)s R X. L Remember that the population standard deviation σ is simply the (n 1)s X R < σ < (n 1)s X L Constructing a Confidence Interval for a Variance and Standard Deviation. First the bad news; there is no magic calculator button to do this for you. 1) Verify that the population has a distribution. ) Identify the sample statistic and the of. of the variance. 3) Find the point estimate s. s (x x ). Many times the will be given to you. n 1 4) Find the critical values X R and X L that correspond to the given level of confidence c. Use Table 6 in Appendix B. 5) Find the left and right endpoints and form the confidence interval for the population. 6) Find the confidence interval for the population standard deviation by taking the of each endpoint. EXAMPLE 1 (Page 345) Find the critical values X R and X L for a 90% confidence interval when the sample size is 0. Because n 0, the degrees of freedom. X R 1 c 1.9 X L 1+c 1+.9 Look in Table 6 in Appendix B (or on the hand-out I gave you) Find the row for the d.f. and the columns for and. X L and X R Now find the critical values X R and X L for a 95% confidence interval when the sample size is 5. d.f.. X R 1.c 1.95
X L 1+c 1+.95 Look in Table 6 in Appendix B (or on the hand-out I gave you) Find the row for the d.f. and the columns for and. X L and X R EXAMPLE (Page 347) You randomly select and weigh 30 samples of an allergy medicine. The sample standard deviation is 1. milligrams. Assuming the weights are normally distributed, construct 99% confidence intervals for the population variance and standard deviation. d.f.. X R 1.c X L 1+c Look in Table 6 in Appendix B (or on the hand-out I gave you) Find the row for the d.f. and the columns for and. X L and X R (n 1)s X < σ < (n 1)s R X, < σ <, < σ < L Now that you have the interval for the variance, take the of the endpoints to find the interval for the.. 0.80 < σ < 3.18, < σ < We are 99% confident that the actual population variance of weights of allergy medicines is between and, and that the actual population standard deviation of weights of allergy medicines is between and. Find the 90% and 95% confidence intervals for the population variance and standard deviation of the medicine weights. d.f.. X R 1.c 1.90 (n 1)s X R X L 1+c 1+.90 X L and X R < σ < (n 1)s X L, < σ <, < σ < ; < σ < d.f.. X R 1.c 1.95 X L 1+c 1+.95 X L and X R (n 1)s X < σ < (n 1)s R X, < σ <, < σ < ; < σ < L