Microsoft Excel was used on a set of data involving the number of defective items found in a random sample of 46 cases of light bulbs produced during a morning shift at a plant. A manager wants to know if the mean number of defective bulbs per case is greater than 20 during the morning shift. She will make her decision using a test with a level of significance of 0.10. The following information was extracted from the Microsoft Excel output for the sample of 46 cases:

n=46, Arithmetic mean=28.00, Std Dev =25.92, standard error=3.82, Null hypothesis: H0 : u<=20, alpha =0.10, df=45, t-test statistic=2.09, one tail test upper critical value =1.3006, p-value=0.021.

i) What parameter is the manager interested in?
ii) State the alternative hypothesis for this study.
iii) What critical value should the manager use to determine the rejection region?
iv) Explain if the null hypothesis should be rejected and why or why not?
v) Explain our risk of committing of a type1 error.

Check Explanation

Step-by-step explanation:

i) The manager hopes to investigate whether there's significant evidence to conclude that the mean number of defective bulbs per case is greater than 20 during the morning shift by comparing the p-value obtained with the significance level at which the hypothesis test was performed at.

ii) The null hypothesis as stated in the question is that the mean number of defective bulbs per case is less than or equal to 20.

The alternative hypothesis takes the other side of the hypothesis; that there is indeed a significant evidence to suggest that the mean number of defective bulbs per case is greater than 20.

Mathematically,

The null hypothesis

H₀: μ₀ ≤ 20

The alternative hypothesis

Hₐ: μ₀ > 20

iii) The critical value the manager should use to determine the rejection region is the significance level at which the test was carried out, that is, significance level of 0.10.

iv) Normally, if the p-value is greater than the significance level, we fail to reject the null hypothesis.

Conversely, if the p-value is less than the significance level, then, we reject the null hypothesis.

For this question,

p-value = 0.021

significance level = 0.10

0.021 < 0.1

p-value < significance level

Hence, we can reject the null hypothesis because the p-value is less than the significance level. We can conclude that there is enough evidence to suggest that the mean number of defective bulbs per case is greater than 20.

e) A type 1 error occurs when we reject the null hypothesis, when in reality, the null hypothesis is true.

Now that we have signified the existence of 3vidence to suggest that the mean number of defective bulbs per case is greater than 20, if the reality is that mean number of defective bulbs per case is less than or equal to 20. A type 1 error can then be stated to have been committed in the rejection of the true hypothesis.

Hope this Helps!!