Perform a hypothesis test that the true average Mwanaidi score is more than 70, as well as the corresponding confidence interval. Set alpha = 0.01. sample mean = 74.52, standard deviation = 10.31 and sample size = 100 subjects
To perform a hypothesis test and calculate the corresponding confidence interval, we will follow these steps:
Step 1: State the null hypothesis (H0) and alternative hypothesis (Ha):
- H0: The true average Mwanaidi score is not more than 70 (μ <= 70)
- Ha: The true average Mwanaidi score is more than 70 (μ > 70)
Step 2: Set the significance level (α) to 0.01.
Step 3: Compute the test statistic and p-value:
- Test Statistic (t):
t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))
t = (74.52 - 70) / (10.31 / sqrt(100))
t = 4.52 / (10.31 / 10)
t = 4.52 / 1.031
t ≈ 4.38
- Degrees of Freedom (df) = sample size - 1 = 100 - 1 = 99
- p-value: The p-value can be obtained from the t-distribution table or using statistical software. For t ≈ 4.38 with df = 99, the p-value is significantly less than 0.01 (it may be around 0.00001 or even lower).
Step 4: Make a decision:
Since the p-value (less than 0.01) is less than the significance level α, we reject the null hypothesis. There is sufficient evidence to support the claim that the true average Mwanaidi score is more than 70.
Step 5: Calculate the confidence interval:
To calculate the confidence interval, we can use the formula:
Confidence Interval = sample mean ± (t * (sample standard deviation / sqrt(sample size)))
Using the values given:
Confidence Interval = 74.52 ± (2.626 * (10.31 / sqrt(100)))
Confidence Interval = 74.52 ± (2.626 * 1.031)
Confidence Interval ≈ 74.52 ± 2.706
Confidence Interval ≈ (71.814, 77.226)
Therefore, the 99% confidence interval for the true average Mwanaidi score is estimated to be (71.814, 77.226).