(a) Explanation Calculation: Central limit theorem: The characteristics of the sampling distribution x ¯ , for all the sample size n is determined using the central limit theorem. Based on central limit theorem the sample size n must be greater than or equal to 30 for making the distribution of sample means x ¯ to be approximately normal, even if the population is not normally distributed . Sampling distribution: For x distribution the sampling distribution of x ¯ is, μ x ¯ = μ σ x ¯ = σ n In the formula μ denotes the population mean, σ denotes the population standard deviation, and n denotes the sample size. The x ¯ can be converted into standard normal distribution as, z = x ¯ − μ x ¯ σ x ¯ In the formula, x ¯ is the mean raw score, μ x ¯ is the mean of the sampling distribution and σ x ¯ is the standard deviation of the sampling distribution. The random samples of size n = 49 are considered. Since the sample size is greater than 30. The sampling distribution of x ¯ follows normal distribution based on central limit theorem. The mean of the x distribution is μ = 15 . The sampling distribution has the mean same as the population x distribution. That is, μ x ¯ = μ = 15 Hence, the value of μ x ¯ is 15. The standard deviation of the distribution is, σ x ¯ = 14 49 = 14 7 = 2 Hence, the value of σ x ¯ is 2 (b) To determine Find the value of μ x ¯ . Find the value of σ x ¯ . Find the value of P ( 15 ≤ x ¯ ≤ 17 ) . (c) To determine Explain why the probability of part (b) is expected to be higher than part (a).

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Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Chapter 6.5, Problem 11P

(a)

To determine

Find the value of μx¯.

Find the value of σx¯.

Find the value of P(15≤x¯≤17).

(b)

To determine

Find the value of μx¯.

Find the value of σx¯.

Find the value of P(15≤x¯≤17).

(c)

To determine

Explain why the probability of part (b) is expected to be higher than part (a).

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Chapter 6 Solutions

Bundle: Understandable Statistics: Concepts And Methods, 12th + Jmp Printed Access Card For Peck's Statistics + Webassign Printed Access Card For ... And Methods, 12th Edition, Single-term

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Find the value of μ x ¯ . Find the value of σ x ¯ . Find the value of P ( 15 ≤ x ¯ ≤ 17 ) .

Solution Summary: The author calculates the sampling distribution using the central limit theorem. The sample size n must be greater than or equal to 30 for the distribution to be approximately normal, even if the population is not normally distributed.

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Find the value of μx¯. Find the value of σx¯. Find the value of P(15≤x¯≤17).

Find the value of μx¯. Find the value of σx¯. Find the value of P(15≤x¯≤17).

Explain why the probability of part (b) is expected to be higher than part (a).

Chapter 6 Solutions

Find the value of μx¯.

Find the value of σx¯.

Find the value of P(15≤x¯≤17).

Find the value of μx¯.

Find the value of σx¯.

Find the value of P(15≤x¯≤17).