Question

A simple random sample of size n is drawn from a population that is normally distributed. The sample mean, x, is found to be 50, and the sample standard deviation, s, is found to be 8.

a) Describe the sampling distribution of the sample mean x.

b) Construct a 98% confidence interval for μ if the sample size, n, is 20.

c) Construct a 98% confidence interval for μ if the sample size, n, is 15. How does decreasing the sample size affect the margin of error, E?

d) Construct a 95% confidence interval for μ if the sample size, n, is 20. How does decreasing the level of confidence affect the margin of error, E?

e)If the population had not been normally distributed could we have computed the confidence intervals in parts (b) – (d)?

Answer 1

a)
Given
Population mean = 50
Sampling distributon of Sample Mean
Sample size (n) = 20
Standard deviation (s) = 8
Confidence interval(in %) = 98

b)

Required confidence interval = (50.0-4.5428, 50.0+4.5428)
Required confidence interval = (45.4572, 54.5428)

c)
Mean = 50
Sample size (n) = 15
Standard deviation (s) = 8
Confidence interval(in %) = 98

Required confidence interval = (50.0-5.4211, 50.0+5.4211)
Required confidence interval = (44.5789, 55.4211)
As the sample size decreases​, the margin of error (E) increases.

d)
Mean = 50
Sample size (n) = 20
Standard deviation (s) = 8
Confidence interval(in %) = 95


Required confidence interval = (50.0-3.7441, 50.0+3.7441)
Required confidence interval = (46.2559, 53.7441)

As the the level (percent)of confidence decreases, Margin of error decrases and simultaneously, the size of the interval decreases.

e)
No, the population needs to be normally distributed.

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