Question

A sample of n = 15 observations is taken from a normal population with unknown mean and variance. The sample has a mean of ¯x = 88.1 and standard deviation 14.29. The researcher hypothesizes that (a) the population has a mean of 90 and (b) the population has a variance of 300. Using confidence intervals, determine the validity of the researcher’s hypotheses at an overall confidence level of 95% assuming the two confidence intervals are independent. That is, recalling that the confidence level is (1−α)100% where α is the significance level, determine the value of α needed so that the overall type I error is 5%. Clearly explain all your work, and discuss the validity of the researcher’s hypotheses in light of the provided data and constructed confidence intervals.

Answer 1

(a) 95% Confidence Interval for the Mean, Unknown Population Standard Deviation

Given : = 88.1, s = 14.29, n = 15, = 0.05

Since population standard deviation is known but n < 30, the tcritical (2 tail) for = 0.05, for df = n -1 = 14, is 2.145

The Confidence Interval is given by ME, where

The Lower Limit = 88.1 - 7.92 = 80.18

The Upper Limit = 88.1 + 7.92 = 96.02

The 95% CI is (80.18, 96.02)

The researchers Hypothesis the the population mean is equal to 90 is valid (We would fail to reject the null Hypothesis), as the value of 90 lies within the limits of the confidence interval.

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(b) 95% Confidence Interval for the Variance

From the given data we calculate s² = 204.2041, n = 15, and degrees of freedom = n - 1 = 14. = 0.05

We find critical values for /2 = 0.025and 1- /2 = 1 – 0.025 = 0.975, df = 14

The lower critical value and

the upper critical critical value .

The confidence interval is given by:

The researchers Hypothesis that the population variance is equal to 300 is valid (We would fail to reject the null Hypothesis), as the value of 300 lies within the limits of the confidence interval

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