You are wondering whether male or female students pay more for books and materials at the U of C. You find 5 male friends at the U of C and find that they spent an average of $680 with a standard deviation of $5. You find 4 female friends at the U of C and find that they spent an average of $690 with a standard deviation of $7. Is there a statistically significant difference in the average amount spent between the male and female friends (at α = .05)?
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to your Z score.
Imagine that the probability of answering this question correctly is only p = .4, and on tests in which it is used, only 8 people attempt it. On what proportion of such tests will fewer than 3 people answer the question correctly?
To determine whether there is a statistically significant difference in the average amount spent between the male and female friends, we can conduct a t-test. Here's how you can calculate and interpret the results:
1. Set up the hypotheses:
- Null hypothesis (H0): There is no difference in the average amount spent between male and female friends at the U of C.
- Alternative hypothesis (Ha): There is a difference in the average amount spent between male and female friends at the U of C.
2. Determine the significance level (α): In this case, α = 0.05.
3. Calculate the t-value:
- First, calculate the degrees of freedom (df) using the formula: df = n1 + n2 - 2.
- n1 = number of males = 5
- n2 = number of females = 4
- df = 5 + 4 - 2 = 7
- Next, calculate the pooled standard deviation (Sp) using the formula:
- Sp = sqrt(((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2))
- s1 = standard deviation of male friends = $5
- s2 = standard deviation of female friends = $7
- Sp = sqrt(((5 - 1) * 5^2 + (4 - 1) * 7^2) / (5 + 4 - 2))
= sqrt((4 * 25 + 3 * 49) / 7) = sqrt((100 + 147) / 7) = sqrt(24.71) ≈ 4.97
- Finally, calculate the t-value using the formula:
- t = (x1 - x2) / (Sp * sqrt(1/n1 + 1/n2))
- x1 = average amount spent by male friends = $680
- x2 = average amount spent by female friends = $690
- t = (680 - 690) / (4.97 * sqrt(1/5 + 1/4)) ≈ -0.20
4. Determine the critical t-value:
- Look up the critical value from the t-distribution table or use statistical software. In this case, with α = 0.05 and df = 7, the critical t-value is approximately ±2.365 (two-tailed test).
5. Compare the calculated t-value with the critical t-value:
- Since |-0.20| < 2.365, the calculated t-value is not greater than the critical t-value.
- We fail to reject the null hypothesis.
6. Interpretation of results:
- There is not enough evidence to suggest a statistically significant difference in the average amount spent between male and female friends at the U of C at α = 0.05.
Note: It's worth mentioning that the sample sizes in this scenario are relatively small, which may limit the generalizability of the results to the broader population of U of C students.