Question

USA Today reports that the average expenditure on Valentine's Day was expected to be $100.89. Do male and female consumers differ in the amounts they spend? The average expenditure in a sample survey of 58 male consumers was $139.46, and the average expenditure in a sample survey of 37 female consumers was $68.15. Based on past surveys, the standard deviation for male consumers is assumed to be $35, and the standard deviation for female consumers is assumed to be $12. The z value is 2.576.

Answer 1

To determine if male and female consumers differ in the amounts they spend on Valentine's Day, we can perform a hypothesis test using the provided information.

Null Hypothesis (H₀): There is no difference in the amounts spent by male and female consumers.

Alternative Hypothesis (H₁): There is a difference in the amounts spent by male and female consumers.

We'll conduct an independent samples t-test to compare the means of the two samples.

Given:

Sample size of male consumers (n₁) = 58

Sample size of female consumers (n₂) = 37

Mean expenditure of male consumers (x₁) = $139.46

Mean expenditure of female consumers (x₂) = $68.15

Standard deviation of male consumers (σ₁) = $35

Standard deviation of female consumers (σ₂) = $12

Z-value = 2.576

We can calculate the test statistic using the formula:

t = (x₁ - x₂) / sqrt((s₁²/n₁) + (s₂²/n₂))

Where s₁ and s₂ are the sample standard deviations.

Substituting the values:

t = ($139.46 - $68.15) / sqrt(($35²/58) + ($12²/37))

Calculating this expression:

t ≈ 71.31 / sqrt(20.34 + 4.32)

t ≈ 71.31 / sqrt(24.66)

t ≈ 71.31 / 4.966

t ≈ 14.36

The calculated t-value is approximately 14.36.

Next, we'll compare the calculated t-value with the critical t-value at a significance level (α) of 0.05 (assuming a two-tailed test). Since the sample sizes are large, we can assume normality and use the t-distribution.

The degrees of freedom (df) can be calculated using the formula:

df = (s₁²/n₁ + s₂²/n₂)² / [((s₁²/n₁)² / (n₁ - 1)) + ((s₂²/n₂)² / (n₂ - 1))]

Substituting the values:

df = ((35²/58) + (12²/37))² / [((35²/58)² / (58 - 1)) + ((12²/37)² / (37 - 1))]

Calculating this expression:

df = (1225/58 + 144/37)² / [(1225²/58² / 57) + (144²/37² / 36)]

df ≈ 52.50

Using statistical software or a t-table, we find that the critical t-value for a two-tailed test with α = 0.05 and df = 52 is approximately 2.009.

Since the calculated t-value (14.36) is greater than the critical t-value (2.009), we reject the null hypothesis. This suggests that there is a significant difference in the amounts spent by male and female consumers on Valentine's Day.

Therefore, based on the provided data, male and female consumers differ in the amounts they spend on Valentine's Day.