Question

Suppose a long jumper claims that his jump distance is not equal to 16 feet, on average. He decides to do a hypothesis test at a 10% significance level to persuade his teammates. The mean distance of the sample jumps is 15.4 feet, with a standard deviation of 1.8 feet. Should he reject the claim? True False

Answer 1

The student aims to determine if their average long jump distance is not equal to 16 feet using a hypothesis test. To conclude whether to reject the null hypothesis, we need the sample size to calculate the test statistic and p-value, which are not provided. Therefore, we cannot establish if the claim should be rejected without additional information.

Step-by-step explanation:

The student is performing a hypothesis test to determine if their average long jump distance is not equal to 16 feet. In this case, we want to test if the true mean is different from 16 feet, so our null hypothesis (H0) is μ = 16, and the alternative hypothesis (Ha) is μ ≠ 16. Since we're checking for inequality, this indicates a two-tailed test.

The sample has a mean of 15.4 feet and a standard deviation of 1.8 feet. However, the data provided does not give us a sample size, which we need to calculate the test statistic and p-value. Assuming that the student has the sample size, the following steps should be taken:

Use the sample mean, population mean, sample standard deviation, and sample size to compute the test statistic.

Determine the critical value at a 10% significance level for a two-tailed test.

Compare the test statistic to the critical value(s).

If the test statistic falls within the critical region, reject the null hypothesis. Otherwise, do not reject the null hypothesis.

Without the sample size and the actual calculation of the test statistic and corresponding p-value, we cannot definitively say whether the long jumper should reject the claim that the average jump distance is 16 feet.

Answer 2

False. Based on the information provided, the long jumper should not reject his claim at a 10% significance level.

Here's why:

1. Hypothesis testing framework:

Null hypothesis (H0): The long jumper's average jump distance is equal to 16 feet.

Alternative hypothesis (Ha): The long jumper's average jump distance is not equal to 16 feet.

Significance level (α): 10% chance of falsely rejecting the null hypothesis when it's true.

2. Test statistic calculation:

We need to calculate the z-score, which measures how many standard deviations the mean jump distance (15.4 ft) is away from the hypothesized mean (16 ft).

z = (15.4 - 16) / 1.8 ≈ -0.33.

3. Critical values and decision rule:

At a 10% significance level, the critical z-scores are +/- 1.645. This means if the calculated z-score falls outside this range, we reject the null hypothesis.

In this case, -0.33 is well within the range of non-rejection (-1.645 to 1.645).

4. Conclusion:

Since the calculated z-score does not fall outside the critical region, we fail to reject the null hypothesis.

This means there is not enough evidence to conclude, at a 10% significance level, that the long jumper's average jump distance is different from 16 feet.

Therefore, the long jumper cannot claim to have statistically proven that his average jump distance is different from 16 feet based on this sample data and significance level. He should further investigate or collect more data before drawing stronger conclusions.