Question

A manufacturer is interested in estimating the lifetime of printers. A confidence interval is constructed based on a sample of 20 printers and the assumption that the population distribution of the lifetime of a printer is normal.

1. If the population standard deviation is 1.5 and the upper end of a 95% confidence interval is 6.36, find the observed sample mean (show your final answer correct to two decimal places).
2. If, instead, we only know the sample standard deviation (which is 1.2), and given the lower end of a 90% confidence interval (which is 3.14), find the observed sample mean (show your final answer correct to two decimal places).
3. Given the significance level β = 0.1, and the sample standard deviation is s = 1.2, perform a hypothesis testing (with detailed steps) to test if the population variance ?² = 1.8.

Answer 1

1. UL = x + z*(σ/√n)

6.36 = x + 1.96*(1.5/√20)

x = 6.36 - 1.96*(1.5/√20) = 5.703

2. LL = x - t*(s/√n)

3.14 = x - 1.73*(1.2/√20)

x = 3.14 + 1.73*(1.2/√20) = 3.604

3. The hypothesis being tested is:

Null hypothesis	H₀: σ² = 1.8
Alternative hypothesis	H₁ : σ² ≠ 1.8

Method	Test Statistic	DF	P-Value
Chi-Square	15.20	19	0.580

The p-value is 0.580.

Since the p-value (0.580) is greater than the significance level (0.10), we fail to reject the null hypothesis.

Therefore, we can conclude that ?² = 1.8.

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