The average weight of a banana is 100 grams. An agricultural scientist buys bananas from a supermarket. Their weights, in grams, are as follows:

\[ w \leftarrow c(102.5, 102.9, 99.9, 101.9, 101.3, 98.9, 96.5, 97.7, 101.8, 100.3) \]

Standard Deviation: 0.9235662
Mean value: 10.8819

She suspects that this sample of bananas is heavier than average and wonders if this supermarket is selling bananas that are heavier than the notional 100g mean weight.

a. State a sensible null hypothesis.

b. State the precise definition of p-value and explain what "more extreme" means in this context.

c. Is a one-sided or two-sided test needed? Justify your choice.

d. Perform a Student's t-test using R and interpret the results.

e. Give a 95% and 99% confidence interval for the mean.

f. For bonus credit, perform a Z-test and account for any differences you find.

This question pertains to performing a t-test to determine if bananas sold by a supermarket are heavier than the notional mean weight of 100g. It also involves interpreting p-values and confidence intervals. A sensible null hypothesis is stated, and R is used to perform the t-test and calculate the confidence intervals.

Explanation:

a. A sensible null hypothesis would be that the mean weight of bananas from this supermarket is equal to the notional 100g mean weight.

b. The p-value is a measure of the strength of evidence against the null hypothesis. In this context, 'more extreme' means values that are farther away from the notional mean weight, both on the heavier and lighter side.

c. A one-sided test is needed because the researcher is only interested in whether the bananas are heavier than the notional mean weight.

d. To perform a student t-test using R, you can use the t.test() function and pass the weight data as the argument. The result will provide the test statistic, p-value, and the conclusion about the hypothesis.

e. A 95% confidence interval for the mean weight can be calculated using the t.test() function in R, and it will provide a range within which the true mean weight is likely to fall. Similarly, a 99% confidence interval can be obtained by specifying the 'conf.level' parameter in the t.test() function.

f. To perform a Z test, you would need to know the true population standard deviation, which is not given in this question. Therefore, a Z test is not appropriate in this case.

Learn more about T-tests in Hypothesis Testing here:

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