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Question 1

Using the results $\bar{x} = 80.75$ , $s = 13.01$ , and $n = 12$ , calculate the $t$ -statistic for testing $H_0: \mu = 90$ against $H_1: \mu \neq 90$ .

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Question 2

A researcher is investigating the weights of a certain variety of banana. A random sample of $12$ bananas is selected and their weights, in grams, are recorded as follows:

$142, 138, 150, 145, 133, 141, 148, 139, 144, 140, 146, 135$

The researcher intends to perform a one-sample $t$ -test to determine if the mean weight of these bananas differs significantly from a target value.

List the assumptions required for a valid one-sample $t$ -test and comment on whether they are reasonable for this study.

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Question 3

For a two-tailed one-sample $t$ -test with $n = 12$ at a significance level of $\alpha = 0.05$ , find the critical $t$ -value(s).

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Question 4

Construct a 95% confidence interval for the true mean weight of bananas given $\bar{x} = 80.75$ , $s = 13.01$ , and $n = 12$ .

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Question 5

A new sample of $20$ bananas has $\bar{x} = 88$ and $s = 12$ . Perform a two-tailed one-sample $t$ -test at the $5\%$ significance level ( $\alpha = 0.05$ ) to test $H_0: \mu = 90$ . Compute the $t$ -statistic and state your conclusion.

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Question 6

A two-tailed hypothesis test is performed for $H_0: \mu = 90$ at a significance level of $\alpha = 0.01$ .

The test statistic is found to be $t = -2.464$ with $11$ degrees of freedom.

Determine the conclusion of the test, justifying your answer.

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Question 7

State the null and alternative hypotheses for a one-sample $t$ -test to determine whether the average weight of bananas differs from 90 grams.

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Question 8

The weights of a sample of bananas are recorded to the nearest gram.

Given the banana weights (in grams): 80, 100, 60, 70, 75, 86, 75, 91, 84, 84, 63, 101, calculate the sample mean $\bar{x}$ and sample standard deviation $s$ .

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Question 9

A researcher is testing the mean weight of a certain variety of bananas, $\mu$ . The null hypothesis is $H_0 : \mu = 90\text{ g}$ and the alternative hypothesis is $H_1 : \mu \neq 90\text{ g}$ .

Explain the meaning of the significance level $\alpha=0.05$ and describe a Type I error in the context of this test.

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Question 10

Hypothesis testing for a population mean

State the null and alternative hypotheses for a one-sample $t$ -test to determine whether the average weight of bananas is less than $90$ grams.

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Question 11

For the t-statistic $t = -2.464$ with $df = 11$ , calculate the two-tailed p-value.

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Question 12

A food quality inspector is investigating a claim that the mean weight of a banana in a shipment is $90\text{ grams}$ . A random sample of $12$ bananas is taken and a two-tailed $t$ -test is performed at the $5\%$ significance level.

The null and alternative hypotheses are: $H_0: \mu = 90$ $H_1: \mu \neq 90$

Using the $t$ -statistic $t = -2.464$ and the critical values $\pm 2.201$ , state the conclusion of the test.

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Question Type 9: Initiating the one sample t-test with the hypothesis and critical region Exercises