Question

1. At the Hawaii Pineapple Company, managers are interested in the size of the pineapples grown in the company’s fields. Last year, the weight of the pineapples harvested from one large field was roughly normally distributed with a mean of 31 ounces and a standard deviation of 4 ounces. A different irrigation system was installed in this field after the growing season. Managers wonder if the the mean weight of pineapples grown in the field this year will be different from last. a. Write out the null Ho and alternative hypotheses Ha in terms of the population mean µ. b. If the managers choose to use a significance level of 0.10 and assume σ = 4, identify their power to detect a mean increase of 2 ounces (µa = 33) if they look at a sample that is 30 pineapples this year and use the two-sided alternative. Also identify the probability of making a type 2 error with true µa = 33. c. Draw pictures of the null and alternative distributions of the means and shade the areas that correspond to (i) Type 1 error, (ii) Type 2 error, and (iii) Power from part (b). d. Describe Type 1 and Type 2 errors of the test in context. e. What sample size should the managers use to ensure their 10% level 2-sided test has power of at least 0.9 to detect a true mean of 33 ounces (assuming σ = 4)? Suppose the managers collect a random sample of size 51 from their field this year to test the hypotheses: Ho : µ = 31 versus Ha : µ 6= 31 at a 10% significance level. They still believe it is reasonable to assume the distribution of weights for the pineapples is approximately normal. f. Evaluate whether the assumptions of the t test are reasonably met. g. Compute a t test statistic and p value and draw a conclusion for the hypothesis test Ho : µ = 31 versus Ha : µ 6= 31 at a 10% level. h. Perform a bootstrap hypothesis test of Ho : µ = 31 versus Ha : µ 6= 31 at the 10% level. i. Explain why using the t, z, or bootstrap tools are all reasonable in this scenerio. Pineapple Weights: 33.61, 33.88, 33.11, 37.05, 31.41, 30.69, 33.94, 30.54, 26.04, 36.01, 39.7, 35.6, 34.96, 33.06, 33.63, 34.59, 32.99, 36.08, 32.33, 36.16, 25.06, 35.7, 30.74, 34.33, 40.02, 30.95, 32.68, 36.57, 32.44, 33.58, 26.44, 29.52, 35.39, 37.96, 31.56, 37.47, 34.22, 27.87, 26.08, 30.81, 36.86, 32.92, 34.29, 40.86, 33.3, 28.46, 34.1, 35.85, 37.39, 37.8, 35.36

1. At the Hawaii Pineapple Company, managers are interested in the size of the pineapples grown in the company’s fields. Last year, the weight of the pineapples harvested from one large field was roughly normally distributed with a mean of 31 ounces and a standard deviation of 4 ounces. A different irrigation system was installed in this field after the growing season. Managers wonder if the the mean weight of pineapples grown in the field this year will be different from last.
a. Write out the null Ho and alternative hypotheses Ha in terms of the population mean µ.
b. If the managers choose to use a significance level of 0.10 and assume σ = 4, identify their power to detect a mean increase of 2 ounces (µa = 33) if they look at a sample that is 30 pineapples this year and use the two-sided alternative. Also identify the probability of making a type 2 error with true µa = 33.
c. Draw pictures of the null and alternative distributions of the means and shade the areas that correspond to (i) Type 1 error, (ii) Type 2 error, and (iii) Power from part (b).
d. Describe Type 1 and Type 2 errors of the test in context.
e. What sample size should the managers use to ensure their 10% level 2-sided test has power of at least 0.9 to detect a true mean of 33 ounces (assuming σ = 4)?
Suppose the managers collect a random sample of size 51 from their field this year to test the hypotheses: Ho : µ = 31 versus Ha : µ 6= 31 at a 10% significance level. They still believe it is reasonable to assume the distribution of weights for the pineapples is approximately normal.
f. Evaluate whether the assumptions of the t test are reasonably met.
g. Compute a t test statistic and p value and draw a conclusion for the hypothesis test Ho : µ = 31 versus Ha : µ 6= 31 at a 10% level.
h. Perform a bootstrap hypothesis test of Ho : µ = 31 versus Ha : µ 6= 31 at the 10% level.
i. Explain why using the t, z, or bootstrap tools are all reasonable in this scenerio.

Pineapple Weights:
33.61, 33.88, 33.11, 37.05, 31.41, 30.69, 33.94, 30.54, 26.04, 36.01, 39.7, 35.6, 34.96, 33.06, 33.63, 34.59, 32.99, 36.08, 32.33, 36.16, 25.06, 35.7, 30.74, 34.33, 40.02, 30.95, 32.68, 36.57, 32.44, 33.58, 26.44, 29.52, 35.39, 37.96, 31.56, 37.47, 34.22, 27.87, 26.08, 30.81, 36.86, 32.92, 34.29, 40.86, 33.3, 28.46, 34.1, 35.85, 37.39, 37.8, 35.36

Added by Zachary G.

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