Final answer:
To conduct the hypothesis test, use a one-sample t-test to test the claim that there has been no change in the mean length of the small counterbalance bars. State the null and alternative hypotheses, calculate the test statistic, determine the critical value, compare the test statistic to the critical value, and draw a conclusion based on the comparison.
Step-by-step explanation:
To conduct the hypothesis test, we can use a one-sample t-test. The null hypothesis is that there has been no change in the mean length of the small counterbalance bars, while the alternative hypothesis is that there has been a change. We will use the significance level of 0.02, which means that if the probability of observing the sample mean under the null hypothesis is less than 0.02, we will reject the null hypothesis.
Step 1: State the null and alternative hypotheses:
H0: The mean length of the small counterbalance bars is 43 millimeters (no change).
Ha: The mean length of the small counterbalance bars is not equal to 43 millimeters (there has been a change).
Step 2: Calculate the test statistic:
The test statistic is calculated as follows: t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(n)), where n is the sample size. In this case, the sample mean is the mean length of the small counterbalance bars (43 millimeters), the hypothesized mean is the null hypothesis mean length (43 millimeters), the sample standard deviation is the standard deviation of the small counterbalance bars, and n is the sample size (12).
Step 3: Determine the critical value:
Since the alternative hypothesis is two-sided, we need to find the critical values for a two-tailed test at a significance level of 0.02. We can use the t-distribution table or statistical software to find the critical values.
Step 4: Compare the test statistic to the critical value:
If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 5: Draw a conclusion:
If we reject the null hypothesis, we can conclude that there has been a change in the mean length of the small counterbalance bars. If we fail to reject the null hypothesis, we do not have sufficient evidence to conclude that there has been a change in the mean length.