Construct simple hypothesis tests for means and proportions?

A potato chip company packages its potato chips into 12.0 ounce bags. You find it hard to believe that the bag contains enough potato chips to weigh 12.0 ounces and would like to make an official complaint. Before doing so, you decide to run an experiment so that you can have some confidence that the company’s claim is incorrect. Over the next several months you buy 30 bags of potato chips and weigh the contents of each one. You discover that the mean weight is 11.9 ounces with a standard deviation of 0.4 ounces. You decide that you will only complain if you can be 95% sure that the bags do not contain at least 12.0 ounces of potato chips. You decide to construct a hypothesis test.











Task:











A. Determine if this is a one-tailed or two-tailed test. Justify your decision.











B. State the null hypothesis and alternative hypothesis. Your null hypothesis should assume the company’s claim is correct.











C. Define the term Type I error and explain what a Type I error is in terms of this problem.











D. Define the term level of significance and identify the level of significance for this problem.











E. Calculate the test statistic as a z-score. Show all relevant work.











F. Using a standard table, you determine that the critical value is –1.645. Determine if you are able to reject the null hypothesis and explain how you reached this conclusion. (Your conclusion should include a comment relating the results to the original problem.)

Construct simple hypothesis tests for means and proportions?
Hypothesis Test for mean:





Assuming you have a large enough sample such that the central limit theorem holds, or you have a sample of any size from a normal population with known population standard deviation, then to test the null hypothesis


H0: μ ≤ Δ or


H0: μ ≥ Δ or


H0: μ = Δ


Find the test statistic z = (xbar - Δ ) / (sx / √ (n))





where xbar is the sample average


sx is the sample standard deviation, if you know the population standard deviation, σ , then replace sx with σ in the equation for the test statistic.


n is the sample size





The p-value of the test is the area under the normal curve that is in agreement with the alternate hypothesis.





H1: μ %26gt; Δ; p-value is the area to the right of z


H1: μ %26lt; Δ; p-value is the area to the left of z


H1: μ ≠ Δ; p-value is the area in the tails greater than |z|





If the p-value is less than or equal to the significance level α, i.e., p-value ≤ α, then we reject the null hypothesis and conclude the alternate hypothesis is true.





If the p-value is greater than the significance level, i.e., p-value %26gt; α, then we fail to reject the null hypothesis and conclude that the null is plausible. Note that we can conclude the alternate is true, but we cannot conclude the null is true, only that it is plausible.





The hypothesis test in this question is:





H0: μ ≥ 12 vs. H1: μ %26lt; 12





The test statistic is:


z = ( 11.9 - 12 ) / ( 0.4 / √ ( 30 ))


z = -1.369306





The p-value = P( Z %26lt; z )


= P( Z %26lt; -1.369306 )


= 0.08545176





Since the p-value is greater than the significance level we fail to reject the null hypothesis and conclude μ ≥ 12 is plausible.