To compute the test statistic for whether modified components have a greater mean time between failures than 520 hours, we perform a right-tailed hypothesis test, calculate the test statistic, compare it to the t-distribution, and make a conclusion based on the p-value and the 0.05 significance level.
To compute the value of an appropriate test statistic for the hypothesis test, we first need to state the null hypothesis (H0) and the alternative hypothesis (Ha). Given that we are testing if the mean time between failures for the modified components is greater than 520 hours, we can write:
H0: μ ≤ 520 (The mean time between failures for the modified components is less than or equal to 520 hours)
Ha: μ > 520 (The mean time between failures for the modified components is greater than 520 hours)
This is a right-tailed test. To calculate the test statistic, we use the sample mean (μ), sample standard deviation (s), and sample size (n) from the provided data. Compute the sample mean and sample standard deviation using the data. Then calculate the test statistic using the formula:
test statistic = (sample mean - population mean) / (sample standard deviation/
)
We then compare the test statistic to the t-distribution with n-1 degrees of freedom to find the p-value. If the p-value is less than the significance level (α = 0.05), we reject the null hypothesis in favor of the alternative hypothesis. Otherwise, we do not reject the null hypothesis.
Finally, we make a conclusion based on the decision, such as:
If p-value < α: Reject H0 and conclude that there is sufficient evidence that the mean time between failures is greater than 520 hours.
If p-value ≥ α: Do not reject H0 and conclude that there is insufficient evidence to claim that the mean time between failures is greater than 520 hours.