Normal Probability Distributions

Transcription

1 Normal Probability Distributions Properties of Normal Distributions The most important probability distribution in statistics is the normal distribution. Normal curve A normal distribution is a continuous probability distribution for a random variable,. The graph of a normal distribution is called the normal curve. 1

2 Properties of Normal Distributions Properties of a Normal Distribution 1. The mean, median, and mode are equal.. The normal curve is bell-shaped and symmetric about the mean. 3. The total area under the curve is equal to one. 4. The normal curve approaches, but never touches the - ais as it etends farther and farther away from the mean. 5. Between µ σand µ + σ (in the center of the curve), the graph curves downward. The graph curves upward to the left of µ σand to the right of µ + σ. The points at which the curve changes from curving upward to curving downward are called the inflection points. 3 Properties of Normal Distributions Inflection points Total area = 1 µ 3σ µ σ µ σ µ µ + σ µ + σ µ + 3σ If is a continuous random variable having a normal distribution with mean µ and standard deviation σ, you can graph a normal curve with the equation 1 -( -µ) σ e =.178 π = 3.14 y = e. σ π 4

3 Means and Standard Deviations A normal distribution can have any mean and any positive standard deviation. Inflection points Mean:µ= 3.5 Standard deviation: σ 1.3 The mean gives the location of the line of symmetry. Mean: µ = 6 Inflection points Standard deviation: σ 1.9 The standard deviation describes the spread of the data. 5 Means and Standard Deviations Eample: 1. Which curve has the greater mean?. Which curve has the greater standard deviation? A B The line of symmetry of curve A occurs at = 5. The line of symmetry of curve B occurs at = 9. Curve B has the greater mean. Curve B is more spread out than curve A, so curve B has the greater standard deviation. 6 3

4 Interpreting Graphs Eample: The heights of fully grown magnolia bushes are normally distributed. The curve represents the distribution. What is the mean height of a fully grown magnolia bush? Estimate the standard deviation. µ = 8 The inflection points are one standard deviation away from the mean. σ Height (in feet) The heights of the magnolia bushes are normally distributed with a mean height of about 8 feet and a standard deviation of about 0.7 feet. 7 The Standard Normal Distribution The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The horizontal scale corresponds to z-scores Any value can be transformed into a z-score by using the formula Value- Mean - µ z = =. Standard deviation σ z 8 4

5 The Standard Normal Distribution If each data value of a normally distributed random variable is transformed into a z-score, the result will be the standard normal distribution. The area that falls in the interval under the nonstandard normal curve (the - values) is the same as the area under the standard normal curve (within the corresponding z-boundaries) After the formula is used to transform an -value into a z-score, the Standard Normal Table in Appendi B is used to find the cumulative area under the curve. z 9 The Standard Normal Table Properties of the Standard Normal Distribution 1. The cumulative area is close to 0 for z-scores close to z = The cumulative area increases as the z-scores increase. 3. The cumulative area for z = 0 is The cumulative area is close to 1 for z-scores close to z = 3.49 Area is close to 0. z = z = 0 Area is Area is close to 1. z z =

6 The t-distribution When a sample size is less than 30, and the random variable is approimately normally distributed, it follow a t-distribution distribution. µ t = s n Properties of the t-distribution 1. The t-distribution is bell shaped and symmetric about the mean.. The t-distribution is a family of curves, each determined by a parameter called the degrees of freedom. The degrees of freedom are the number of free choices left after a sample statistic such as is calculated. When you use a t-distribution to estimate a population mean, the degrees of freedom are equal to one less than the sample size. d.f. = n 1 Degrees of freedom Continued. 11 The t-distribution 3. The total area under a t-curve is 1 or 100%. 4. The mean, median, and mode of the t-distribution are equal to zero. 5. As the degrees of freedom increase, the t-distribution approaches the normal distribution. After 30 d.f., the t-distribution is very close to the standard normal z-distribution. d.f. = d.f. = 5 0 Standard normal curve The tails in the t-distribution are thicker than those in the standard normal distribution. t 1 6

7 The Chi-Square Distribution The point estimate for σ is s, and the point estimate for σ is s. s is the most unbiased estimate for σ. If the random variable has a normal distribution, then the distribution of ( n 1) s χ = σ forms a chi-square distribution for samples of any size n > The Chi-Square Distribution Four properties of the chi-square distribution are as follows. 1. All chi-square values χ are greater than or equal to zero.. The chi-square distribution is a family of curves, each determined by the degrees of freedom. To form a confidence interval for σ, use the χ -distribution with degrees of freedom. To form a confidence interval for σ, use the χ -distribution with degrees of freedom equal to one less than the sample size. 3. The area under each curve of the chi-square distribution equals one. 4. Find the critical value z c that corresponds to the given level of confidence. 5. Chi-square distributions are positively skewed. 14 7

8 Critical Values for X There are two critical values for each level of confidence. The value χ R represents the right-tail critical value and χ L represents the left-tail critical value. X R 1 c X X L c ( ) 1 1+ c 1 = X Area to the right of X R Area to the right of X L 1 c X L c X R 1 c The area between the left and right critical values is c. X 15 Critical Values for X Eample: Find the critical values χ R and χ L for an 80% confidence when the sample size is 18. Because the sample size is 18, there are d.f. = n 1 = 18 1 = 17 degrees of freedom, Area to the right of χ R = 1 c 1 = 0.8 = 0.1 Area to the right of χ L = 1 + c 1 + = 0.8 = 0.9 Use the Chi-square distribution table to find the critical values. Continued. 16 8

9 Critical Values for X Eample continued: Appendi B: Table 6: χ -Distribution Degrees of freedom α χ R = χ L = Sampling Distributions and the Central Limit Theorem 9

10 Sampling Distributions A sampling distribution is the probability distribution of a sample statistic that is formed when samples of size n are repeatedly taken from a population. Population 19 Sampling Distributions If the sample statistic is the sample mean, then the distribution is the sampling distribution of sample means The sampling distribution consists of the values of the sample means, 1,, 3, 4, 5,

11 Properties of Sampling Distributions Properties of Sampling Distributions of Means 1. The mean of the sample means, µ, is equal to the population mean. µ = µ σ,. The standard deviation of the sample means, is equal to the σ, population standard deviation, divided by the square root of n. σ = The standard deviation of the sampling distribution of the sample means is called the standard error of the mean. σ n 1 Sampling Distribution of Means Eample: The population values {5, 10, 15, 0} are written on slips of paper and put in a hat. Two slips are randomly selected, with replacement. a. Find the mean, standard deviation, and variance of the population. Population µ = 1.5 σ = 5.59 σ = 31.5 Continued. 11

12 Sampling Distribution of Means Eample continued: The population values {5, 10, 15, 0} are written on slips of paper and put in a hat. Two slips are randomly selected, with replacement. b. Graph the probability histogram for the population values. P() Probability Histogram of Population of 0.5 Probability Population values This uniform distribution shows that all values have the same probability of being selected. Continued. 3 Sampling Distribution of Means Eample continued: The population values {5, 10, 15, 0} are written on slips of paper and put in a hat. Two slips are randomly selected, with replacement. c. List all the possible samples of size n = and calculate the mean of each. 5, 5 5, 10 5, 15 5, 0 10, 5 10, 10 10, 15 10, 0 mean, , 5 15, 10 15, 15 15, 0 0, 5 0, 10 0, 15 0, 0 mean, These means form the sampling distribution of the sample means. Continued. 4 1

13 Sampling Distribution of Means Eample continued: The population values {5, 10, 15, 0} are written on slips of paper and put in a hat. Two slips are randomly selected, with replacement. d. Create the probability distribution of the sample means. f Probability Probability Distribution of Means 5 Sampling Distribution of Means Eample continued: The population values {5, 10, 15, 0} are written on slips of paper and put in a hat. Two slips are randomly selected, with replacement. e. Graph the probability histogram for the sampling distribution. Probability P() 5 Probability Histogram of Sampling Distribution mean The shape of the graph is symmetric and bell shaped. It approimates a normal distribution. 6 13

14 The Central Limit Theorem If a sample of size n 30 is taken from a population with any type of distribution that has a mean = µ and standard deviation = σ, µ µ the sample means will have a normal distribution. µ 7 The Central Limit Theorem If the population itself is normally distributed, with mean = µ and standard deviation = σ, µ the sample means will have a normal distribution for any sample size n. µ 8 14

15 The Central Limit Theorem In either case, the sampling distribution of sample means has a mean equal to the population mean. µ = µ Mean of the sample means The sampling distribution of sample means has a standard deviation equal to the population standard deviation divided by the square root of n. σ = σ n This is also called the standard error of the mean. Standard deviation of the sample means 9 The Mean and Standard Error Eample: The heights of fully grown magnolia bushes have a mean height of 8 feet and a standard deviation of 0.7 feet. 38 bushes are randomly selected from the population, and the mean of each sample is determined. Find the mean and standard error of the mean of the sampling distribution. Standard deviation Mean (standard error) µ = µ = 8 σ = σ n 0.7 = 38 = 0.11 Continued

16 Interpreting the Central Limit Theorem Eample continued: The heights of fully grown magnolia bushes have a mean height of 8 feet and a standard deviation of 0.7 feet. 38 bushes are randomly selected from the population, and the mean of each sample is determined. The mean of the sampling distribution is 8 feet,and the standard error of the sampling distribution is 0.11 feet. From the Central Limit Theorem, because the sample size is greater than 30, the sampling distribution can be approimated by the normal distribution µ = 8 σ = Finding Probabilities Eample continued: Find the probability that the mean height of the 38 bushes is less than 7.8 feet. = 8 µ = 0.11 σ P ( < 7.8) n = P ( < 7.8) = P (z < 1.8? ) = z z µ σ = = 0.11 = 1.8 The probability that the mean height of the 38 bushes is less than 7.8 feet is