Math Problem Statement

A random sample of 110 married or de facto couples in the Sydney region has been selected to study non-verbal communication. An experiment was conducted requiring the subject to guess words shown to the partner, without any verbal communication. Some of the variables recorded for each couple are listed below.

Variable Description subid Subject ID sex sex of guesser years the length of the relationship (in years) percentage the percentage of correct guesses 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 women and men in the average relationship length?’ Which test is correct to use, if we do not know the population standard deviation? Answer 1 Question 2

What is the null hypothesis? Answer 2 Question 2

The p-value of the test is 0.284. What is the decision of the test? Answer 3 Question 2

Part 2 A test is run to answer the research question, ’Is the average relationship length equal to 38?’ Which test is correct to use, if we know the population standard deviation? Answer 4 Question 2

The test statistic is -0.73. 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 2

The p-value of the test is 0.465. What is the decision of the test? Answer 6 Question 2

Solution

Part 1

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

Answer 1: Since we do not know the population standard deviation, and we are comparing the means of two independent groups (men and women), the correct test to use is the independent samples t-test (also known as a two-sample t-test).

Question: What is the null hypothesis?

Answer 2: The null hypothesis (H0H_0) for this test is: H0:μwomen=μmenH_0: \mu_{\text{women}} = \mu_{\text{men}} This states that there is no difference in the average relationship length between women and men.

Question: The p-value of the test is 0.284. What is the decision of the test?

Answer 3: Since the p-value (0.284) is greater than the common significance level (e.g., α=0.05\alpha = 0.05), we fail to reject the null hypothesis. This means there is not enough evidence to conclude that there is a difference in the average relationship length between women and men.

Part 2

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

Answer 4: If the population standard deviation is known, the correct test to use is the one-sample z-test.

Question: The test statistic is -0.73. 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: The test statistic of -0.73 indicates that the sample mean is 0.73 standard deviations below the hypothesized population mean of 38.

Question: The p-value of the test is 0.465. What is the decision of the test?

Answer 6: Since the p-value (0.465) is greater than the common significance level (e.g., α=0.05\alpha = 0.05), we fail to reject the null hypothesis. This means there is not enough evidence to conclude that the average relationship length is different from 38 years.

Would you like more details or have any questions?

Here are some related questions to expand your understanding:

  1. How do we calculate the test statistic for an independent samples t-test?
  2. What are the assumptions that need to be verified for a t-test to be valid?
  3. How do we interpret a p-value in hypothesis testing?
  4. What are the differences between a z-test and a t-test?
  5. How would the decision change if the p-value was less than 0.05?

Tip: Always check the assumptions of normality and equality of variances when using a t-test, as violating these assumptions can affect the validity of the test results.

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
t-test
z-test
p-value

Formulas

Independent samples t-test: t = (X̄1 - X̄2) / sqrt((s1²/n1) + (s2²/n2))
One-sample z-test: z = (X̄ - μ) / (σ / sqrt(n))
P-value: A measure of the strength of evidence against the null hypothesis

Theorems

Central Limit Theorem
Law of Large Numbers

Suitable Grade Level

College Undergraduate

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