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The inspection division of the Lee County Weights and Measures Department is interested in estimating the...

The inspection division of the Lee County Weights and Measures Department is interested in estimating the actual amount of soft drink that is placed in 2-liter bottles at the local bottling plant of a large nationally known soft-drink company. The bottling plant has informed the inspection division that the standard deviation for 2-liter bottles is 0.05 liter, i.e. the population standard deviation is σ = 0.05. A random sample of one hundred 2-liter bottles obtained from this bottling plant indicates a sample average of 1.99 liters.

1. [5 points] Set up a 95% confidence interval estimate of the true average amount of soft drink in each bottle. What is the margin of error?

2. [4 points] Based on your answer to part 1, can we reject the null hypothesis H0 : µ = 2 vs. HA : µ ≠ 2 at a 95% confidence level? Explain.

3. [5 points] What would be your answer to previous question if n = 121? What about if n = 81? What is the range of the sample size such that the above hypothesis is rejected?

4. [8 points] Suppose now that Lee County requires these measurements in ounces instead (1 liter=33.8 ounces). Call this variable Y , that is Y = 33.8X. (a) [2 points] What is the mean of the sample in ounces? (Recall x¯ = 1.99) (b) [2 points] Whatisthe standard deviation ofthe population in ounces? I.e. whatisσy?(Recall σx =0.05) (c) [4 points] What is the margin of error for a 95% confidence interval for the amount of soft drink in a bottle in ounces? Compare your answer to the one in part 1. Hint: Compare the ratio of the margin of errors, using the one for ounces in the numerator. ∗

5. [8points]Let's go back to liters.Assume that we no longer haveσ but we calculateds=0.048 fromthe sample. (a) [3 points] Construct the test statistic for the hypothesis H0 : µ = 2 vs. HA : µ ≠ 2. (b) [5points] Whatisthe(approximate) criticalt value that we should compare this statistic to if we want to use a 90% confidence? For 95% confidence? For 99% confidence? Do you reject the null hypothesis at any ofthese levels?What educated guess can youmake aboutthe p-value of this test?

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