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Section 5.3

Section 5.3. Normal Distributions: Finding Values. Finding z-scores for a given area. Read the table (#4) in reverse 1. Make a quick sketch and shade the area indicated. 2. Decide if the z-score is < or > zero. 3. Look in the BODY for the area given.

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Section 5.3

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  1. Section 5.3 Normal Distributions: Finding Values

  2. Finding z-scores for a given area • Read the table (#4) in reverse • 1. Make a quick sketch and shade the area indicated. • 2. Decide if the z-score is < or > zero. • 3. Look in the BODY for the area given. • 4. Read the row and column headings to determine the z-score.

  3. Ex: Find the z-score for each. • 1. area = 0.9916 • 2. P47 • 3. 78.5% of the distribution is to the right of the z-score. • 4. 25% of the distribution lies between z and –z.

  4. Finding the x-value for a given z-score • Recall: z = x - µ σ • If you know z, µ, and σ: just sub in and solve for x.

  5. Ex: Find the x value • 1. z = 1.62, µ = 13, σ = 0.39 • 2. z = 0.45, µ = 24, σ = 0.7

  6. Finding specific data value for a given probability • Sketch the curve and shade the probability. • Find the z-score. • Solve for x.

  7. 34. The time spent (in days) waiting for a kidney transplant for people ages 35-49 in a recent year can be approximated by a normal distribution with µ = 1674 days and σ = 212.5 days • A) What waiting time represents the 80th percentile? • B) What waiting time represents the first quartile?

  8. 36. The annual per capita consumption of ice cream (in pounds) in the US can be approximated by a normal distribution. µ = 20.7 lb and σ = 4.2 lb. • A) What is the largest annual per capita consumption of ice cream that can be in the bottom 10% of consumption? • B) Between what two values does the middle 80% of consumption lie?

  9. Section 5.4 Sampling Distributions and the Central Limit Theorem

  10. Sampling Distributions • Sampling Distribution: the probability distribution of a sample statistic that is formed when samples of size n are repeatedly taken from a population. • When sample MEANS are used, it’s called a sampling distribution of sample means.

  11. Properties of sampling distributions of Sample Means • The mean of the sample means is equal to the population mean. • The standard deviation of the sample means is equal to the population standard deviation divided by the square root of the sample size. (This is also called the standard error of the mean.)

  12. Example • #2. Given µ = 150 and σ = 25. Find the mean and the standard deviation of a sampling distribution of sample means with sample size n = 100.

  13. The Central Limit Theorem • 1. If samples of size n, where n > 30, are drawn from any population with mean µ and standard deviation σ, then the sampling distribution of sample means approximates a normal distribution. The greater the sample size n, the better the approximation. • 2. If the population itself is normally distributed, then the sampling distribution of sample means is normally distributed for any size n.

  14. Probability and the CLT

  15. # 14. For a sample of n = 100, find the probability of a sample mean being greater than 24.3 if µ = 24 and σ = 1.25 # 26 The population mean annual salary for flight attendants is $56,275. A random sample of 48 flight attendants is selected from this population. What is the probability that the mean annual salary of the sample is less than $56,100? Assume σ = $1800.

  16. # 38. A brake pad manufacturer claims its brake pads will last for 38,000 miles. You work for a consumer protection agency and you are testing this manufacturer’s brake pads. Assume the life spans of the brake pads are normally distributed. You randomly select 50 brake pads. In your tests, the mean life of the brake pads is 37,650 miles. Assumeσ = 1000 miles. • A) What is the probability that the mean of the sample is 37,650 miles or less?

  17. #38, continued • B) What do you think of the manufacturer’s claim? • C) Would it be unusual to have an individual brake pad last for 37,650 miles? Why or why not?

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