Question

# Suppose that we will randomly select a sample of 79 measurements from a population having a...

Suppose that we will randomly select a sample of 79 measurements from a population having a mean equal to 21 and a standard deviation equal to 8. (a) Describe the shape of the sampling distribution of the sample mean . Do we need to make any assumptions about the shape of the population? Why or why not? Normally distributed ; yes , because the sample size is large . (b) Find the mean and the standard deviation of the sampling distribution of the sample mean . (Round your σx¯ answer to 1 decimal place.) µ 21 σ 0.9 (c) Calculate the probability that we will obtain a sample mean greater than 23; that is, calculate P( > 23). Hint: Find the z value corresponding to 23 by using µ and σ because we wish to calculate a probability about . (Use the rounded standard error to compute the rounded Z-score used to find the probability. Round your answer to 4 decimal places. Round z-scores to 2 decimal places.) P( > 23) (d) Calculate the probability that we will obtain a sample mean less than 20.487; that is, calculate P( < 20.487) (Use the rounded standard error to compute the rounded Z-score used to find the probability. Round your answer to 4 decimal places. Round z-scores to 2 decimal places.) P( < 20.487)

Solution :

Given that ,

= 21

= / n = 8 / 79 = 0.9

c) P( > 23) = 1 - P( < 23)

= 1 - P[( - ) / < (23 - 21) / 0.9 ]

= 1 - P(z < 2.22)

= 1 - 0.9868

= 0.0132

d) P( < 20.487) = P(( - ) / < (20.487 - 21) / 0.9)

= P(z < -0.57)

Using z table

= 0.2843

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