Math Problem Statement

A random sample of 102 teams playing in major international team sporting competitions has been selected to study corporate sponsorship and subsequent sporting success. Some of the variables recorded for each team are listed below.

VariableDescription

subid

Subject ID

sex

sex of players

numSponsors

number of distinct corporate sponsors

percentWin

the percentage of international games won

Address the following questions, assuming as necessary that all test assumptions are verified. Part 1 A test is run to answer the research question, ’Is there any difference between men’s teams and women’s teams in the average number of corporate sponsors?’

Which test is correct to use, if we know the population standard deviation?

Answer 1 Question 3 z-testt-test

What is the null hypothesis?

Answer 2 Question 3 There is no differenceThere is a difference

The p-value of the test is 0.012. What is the decision of the test?

Answer 3 Question 3 Accept H0Reject H0Do not accept H0Do not reject H0

Part 2 A test is run to answer the research question, ’Is the average number of corporate sponsors equal to 38?’

Which test is correct to use, if we do not know the population standard deviation?

Answer 4 Question 3 z-testt-test

The test statistic is -2.44. This means that the difference between the sample mean and the possible population mean, as stated in the null hypothesis (i.e. 38), is:

Answer 5 Question 3 -1.96 times the standard error of the sample mean-1.96 times the standard deviation of the number of corporate sponsors-2.44 times the standard deviation of the number of corporate sponsors-2.44 times the standard error of the sample mean

The p-value of the test is 0.017. What is the decision of the test?

Solution

Part 1

  1. Which test is correct to use, if we know the population standard deviation?

    • Since the population standard deviation is known, the correct test to use is the z-test.

    Answer 1: z-test

  2. What is the null hypothesis?

    • The null hypothesis (H₀) is that there is no difference in the average number of corporate sponsors between men's teams and women's teams.

    Answer 2: There is no difference

  3. The p-value of the test is 0.012. What is the decision of the test?

    • A p-value of 0.012 is less than the typical significance level of 0.05, meaning the test has found statistically significant evidence to reject the null hypothesis. Thus, we should reject H₀.

    Answer 3: Reject H0

Part 2

  1. Which test is correct to use, if we do not know the population standard deviation?

    • When the population standard deviation is unknown, the appropriate test to use is the t-test.

    Answer 4: t-test

  2. The test statistic is -2.44. This means that the difference between the sample mean and the possible population mean, as stated in the null hypothesis (i.e. 38), is:

    • The test statistic of -2.44 indicates that the difference between the sample mean and the hypothesized population mean is -2.44 times the standard error of the sample mean.

    Answer 5: -2.44 times the standard error of the sample mean

  3. The p-value of the test is 0.017. What is the decision of the test?

    • Since the p-value of 0.017 is less than the significance level of 0.05, we would reject H₀ (the null hypothesis that the average number of corporate sponsors is equal to 38).

    Answer 6: Reject H0

Do you need more details or have any questions about these steps? Here are some related questions:

  1. What assumptions are needed for a z-test and t-test?
  2. How do you calculate the p-value from a test statistic?
  3. How does sample size affect the choice between a z-test and t-test?
  4. Why is knowing or not knowing the population standard deviation important for test selection?
  5. What are Type I and Type II errors in hypothesis testing?

Tip: Always check if your test meets the assumptions for the test (like normality or sample size) before interpreting the results!

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
z-Test
t-Test
P-Value
Test Statistic
Standard Error

Formulas

z = (X̄ - μ) / (σ / √n)
t = (X̄ - μ) / (s / √n)

Theorems

Central Limit Theorem
Hypothesis Testing

Suitable Grade Level

University Level

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