| Two-Sample Hypothesis A national manufacturer of ball bearings is experimentin
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Question
| Two-Sample Hypothesis A national manufacturer of ball bearings is experimenting with two different processes for producing precision ball bearings. It is important that the diameters be as close as possible to an industry standard. The output from each process is sampled and the average error from the industry standard is measured in millimeters. The results of the samples are presented next. Process A Process B 3.0 0.5 14 Sample mean 2.0 Standard deviation 1.0 12 Sample size The researcher is interested in determining whether there is evidence that the two processes yield different average errors. The population standard deviations are unknown but are assumed equal. Answer the following Assume you are trying to conclude if the two processes yield different average errors a. Determine if the appropriate hypothesis test should be an upper tail, lower tail, or 2-tail test b. Determine the value of the test statistic c. Determine the critical z or t value using a level of significance of 5% d. Determine the p-value. e. State your conclusion: Reject Ho or Do not Reject Ho? (Using a significance level of 5%) What is the probability of making a Type I Error? f.Explanation / Answer
Solution:-
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: 1 - 2 = 0
Alternative hypothesis: 1 - 2 0
Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.05.. Using sample data, we will conduct a two-sample z-test of the null hypothesis.
Analyze sample data. Using sample data, we compute the standard error (SE), z statistic test statistic (z).
SE = sqrt[(s12/n1) + (s22/n2)]
SE = 0.3181
DF = 24
z = [ (x1 - x2) - d ] / SE
z = - 3.14
zcritical = + 1.96
where s1 is the standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2, x1 is the mean of sample 1, x2 is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.
Since we have a two-tailed test, the P-value is the probability that a z statistic more extreme than -3.14; that is, less than - 3.14 or greater than 3.14.
Thus, the P-value = 0.0016
Interpret results. Since the P-value (0.0016) is less than the significance level (0.05), we cannot accept the null hypothesis.
Reject H0,
The probability of making type I error is 0.05.
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