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Chapter 6:

Chapter 6:. Section 6.5: Central Limit Theorem. Properties of the Distribution of Sample Means. 1. The mean of the sample means will be the same as the population mean. 2. The standard deviation of the sample means will be smaller than the standard deviation of

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Chapter 6:

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  1. Chapter 6: Section 6.5: Central Limit Theorem

  2. Properties of the Distribution of Sample Means 1. The mean of the sample means will be the same as the population mean 2. The standard deviation of the sample means will be smaller than the standard deviation of the population; and standard deviation of the mean = σ √n

  3. In the previous lesson, we learned that when solving for z: X – µ σ z = Now, since we know the standard deviation of The sample means is: σ √n σ √n We simply plug in for the standard deviation In the same equation and use X instead of X: _ _ X – µ z = σ √n

  4. **HINT: When using this equation in the Calculator, be sure to put parenthesis around BOTH the numerator and the denominator _ (X – µ) z = ( ) σ √n

  5. Difference between the z = equations: X – µ σ 1. For individual data: z = _ (X – µ) 2. For sample means: z = σ √n

  6. EXAMPLE 1: The mean time to complete a psychology exam is 34 minutes with a standard deviation of 8. If a class of 30 students takes the exam, find the probability that the sample mean will be more than 30 minutes. Are we dealing with individual data or sample means? _ (X – µ) Sample means, so we use the equation z = σ √n µ = 34 σ= 8 n = 30 _ More than 30 minutes: X = 30 _ (30 – 34) (X – µ) = = -2.74 z = σ √n 8 √30 ( )

  7. EXAMPLE 1: The mean time to complete a psychology exam is 34 minutes with a standard deviation of 8. If a class of 30 students takes the exam, find the probability that the sample mean will be more than 30 minutes. µ = 34 σ= 8 n = 30 z = -2.74 _ More than 30 minutes: X = 30 -2.74 0 z = -2.74 Area = 0.4969 Since the questions asks for MORE THAN 30, shade to the right Area = 0.4969 + 0.5 = 0.9969

  8. EXAMPLE 2: The average hourly wage of fast-food workers employed by a nationwide chain is $5.55. The standard deviation is $1.15. If a sample of 50 workers is selected, find the probability that the mean of the sample will be between $5.25 and $5.90? Are we dealing with individual data or sample means? _ (X – µ) Sample means, so we use the equation z = σ √n µ = 5.55 σ= 1.15 n = 50 _ Between $5.25 and $5.90: X = 5.25, 5.90 (5.90 – 5.55) (5.25 – 5.55) z = z = = 2.15 = -1.84 1.15 √50 1.15 √50 ( ) ( )

  9. EXAMPLE 2: The average hourly wage of fast-food workers employed by a nationwide chain is $5.55. The standard deviation is $1.15. If a sample of 50 workers is selected, find the probability that the mean of the sample will be between $5.25 and $5.90? z = -1.84, 2.15 µ = 5.55 σ= 1.15 n = 50 _ Between $5.25 and $5.90: X = 5.25, 5.90 -1.84 0 2.15 z = -1.84 Area = 0.4671 z = 2.15 Area = 0.4842 Since the questions asks for BETWEEN, shade between and add Area = 0.4671 + 0.4842 = 0.9513

  10. EXAMPLE 3: The mean grade point average of the engineering majors at a large university is 3.23, with a standard deviation of 0.72. In a class of 48 students, find the probability that the mean grade point average of the student is less than 3.15. Are we dealing with individual data or sample means? _ (X – µ) Sample means, so we use the equation z = σ √n µ = 3.23 σ= 0.72 n = 48 _ Less than 3.15: X = 3.15 _ (3.15 – 3.23) (X – µ) = z = = -0.77 σ √n 0.72 √48 ( )

  11. EXAMPLE 3: The mean grade point average of the engineering majors at a large university is 3.23, with a standard deviation of 0.72. In a class of 48 students, find the probability that the mean grade point average of the student is less than 3.15. z = -0.77 µ = 3.23 σ= 0.72 n = 48 _ Less than 3.15: X = 3.15 -0.77 0 z = -0.77 Area = 0.2794 Since the questions asks for LESS THAN, shade to the left Area = 0.5 – 0.2794 = 0.2206

  12. EXAMPLE 4: At a large university, the mean age of graduate students who are majoring in psychology is 32.6 years, and the standard deviation is 3 years. Assume normally distributed. If an individual from the department is randomly selected, find the probability that his/her age is less than 33.2 years. Are we dealing with individual data or sample means? X – µ σ Individual data, so we use the equation z = µ = 32.6 σ= 3 Less than 33.2: X = 33.2 X – µ σ 33.2 – 32.6 3 = 0.20 = z = z = 0.20 Area = 0.0793 Area = 0.5 + 0.0793 = 0.5793 0 0.20

  13. EXAMPLE 4: At a large university, the mean age of graduate students who are majoring in psychology is 32.6 years, and the standard deviation is 3 years. Assume normally distributed. If a random sample of 15 individuals is selected, find the probability that the mean age of the students in the sample Will be less than 33.2 years. _ Are we dealing with individual data or sample means? (X – µ) Sample means, so we use the equation z = σ √n Less than 33.2: _ µ = 32.6 σ= 3 n = 15 _ X = 33.2 (X – µ) (33.2 – 32.6) z = = = 0.77 σ √n ( ) _3 √15 z = 0.77 Area = 0.2794 Area = 0.5 + 0.2794 = 0.7794 0 0.77

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