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In: Statistics and Probability

A retailer discovers that 3 jars from his last shipment of Spiffy peanut butter contained between...

A retailer discovers that 3 jars from his last shipment of Spiffy peanut butter contained between 15.85 and 15.92 oz of peanut butter, despite the labeling indicating that each jar should contain 16 oz. of peanut butter. He is wondering if Spiffy is cheating its customers by filling its jars with less product than advertised. He decides to measure the weight of 50 jars from the shipment and use hypothesis testing to verify this.

(a) What are the null and alternative hypotheses for this experiment?

(b) Describe, in words, a Type I error for this experiment.

(c) Describe, in words, a Type II error for this experiment.

(d) Given the answer to (a), should the null hypothesis be rejected when the sample mean falls below or over a certain threshold? Should this threshold be below or above the value 16.0 oz?

(e) What is the distribution of X ̄, the sample mean?

(f) In his sample of 50 jars, the retailer finds an average weight of 15.84 oz and a sample standard deviation of 0.5 oz. He decides to use a significance level of 0.04. What is the conclusion from this hypothesis testing? Can you conclude that Spiffy is cheating its customers?

(g) What is the p-value? What is the meaning of this number?

(h) For what values of the sample mean would the null hypothesis be rejected?

(i) Calculate the probability of type II error if the true mean is 15.7 oz.

(j) Solve (f), (h) and (i) when the level of significance is 0.01. Is your new answer for (f) consistent with the p-value found in (g)? How is the probability of type II error affected when the probability of type I error changes?

Solutions

Expert Solution

a) The hypotheses are:

b) Type I error occurs when we reject correct null hypothesis. Here, type I error occurs when we conclude that the average weight of jars is 16 oz but actually Spiffy is cheating its customers by filling its jars with less product than advertised.

c) Type II error occurs when we fail to reject incorrect null hypothesis. Here, type II error occurs when we conclude that the average weight of jars is less 16 oz but actually Spiffy is NOT cheating its customers by filling its jars with less product than advertised.

d) The null hypothesis should be rejected when the sample mean falls below a certain threshold because this is left tailed test.

orchestra answered 2 years ago
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