The 95% confidence intervals are (9.244, 9.756) for females and (8.687, 9.313) for males. There is no evidence to suggest that female gymnasts have a higher population mean score than male gymnasts at the 5% significance level. Assessing the assumptions of independent samples and normality requires the normal quantile plots. While homogeneity of variances is unlikely, adjustments can be considered.
a) Confidence Intervals for Population Means:
Female Gymnasts:
Sample mean: 9.5
Sample standard deviation: 1.2
Sample size: 25
We need to calculate the critical value for a 95% confidence interval with 24 degrees of freedom (n - 1). Using a t-distribution table or calculator, the critical value is approximately 2.064.
Margin of error: 2.064 * (1.2 / √25) = 0.256
Confidence interval: (9.5 - 0.256, 9.5 + 0.256) = (9.244, 9.756)
2.Male Gymnasts:**
Sample mean: 9.0
Sample standard deviation: 1.5
Sample size: 30
We need to calculate the critical value for a 95% confidence interval with 29 degrees of freedom (n - 1). Using a t-distribution table or calculator, the critical value is approximately 2.045.
Margin of error: 2.045 * (1.5 / √30) = 0.313
Confidence interval: (9.0 - 0.313, 9.0 + 0.313) = (8.687, 9.313)
b) Hypothesis Testing for the Difference in Population Means:
Null Hypothesis (H_0):** There is no difference in the population mean floor exercise scores between female and male gymnasts.
Alternative Hypothesis (H_a):The population mean floor exercise score for female gymnasts is higher than that of male gymnasts.
Significance level (α):** 5%
We will use the two-tailed t-test for independent samples with unequal variances.
1. Calculate the pooled standard error:
Pooled variance = [(25 - 1) * 1.2^2 + (30 - 1) * 1.5^2] / (25 + 30 - 2) = 1.767
Pooled standard error = √(1.767 / 25 + 1.767 / 30) = 0.271
2. Calculate the t-statistic:
t = (9.5 - 9.0) / 0.271 = 1.845
3. Determine the critical value:
Degrees of freedom = 53 (smallest sample size - 1)
Critical value for a two-tailed test with α = 0.05 and 53 degrees of freedom is approximately 2.012.
4. Compare the t-statistic with the critical value:
Since 1.845 < 2.012, we fail to reject the null hypothesis.
Based on the hypothesis test, there is not enough evidence to suggest that female gymnasts have a higher population mean floor exercise score than male gymnasts at the 5% significance level.
complete the question
In the Female and Male Apparatus Finals, the floor exercise scores were analyzed. For the Female Finals, the sample mean floor exercise score was 9.5 with a sample standard deviation of 1.2, based on 25 gymnasts' performances. In the Male Finals, the sample mean floor exercise score was 9.0 with a sample standard deviation of 1.5, calculated from 30 gymnasts' performances.
a) Calculate the 95% confidence intervals for the population mean floor exercise scores for both female and male gymnasts.
b) Perform a hypothesis test at the 5% significance level to determine if there's evidence to suggest that female gymnasts have a higher population mean floor exercise score in the Apparatus Finals than male gymnasts. Use the t-test for the difference of population means.
c) Discuss the assumptions necessary for the validity of the t-test in this scenario and assess if these assumptions are reasonably met based on the provided sample sizes and the normality of data, which is depicted in the normal quantile plots below for female and male floor exercise scores.