a random sample of 80 college students was selected to participate in a experiment. the mean of the sample is 317 points, the standard deviation of 37 there are 310 students that didn't complete the experiment
1. compute the appropriate test statistic?
2. what is the critical value of the test statistic?
3. given a data, what would you conclude
4. calculate a 95% and 99% confidence interval?
What hypothesis are you testing?
To answer these questions, we will need to assume that the population follows a normal distribution.
1. To compute the appropriate test statistic, we can perform a hypothesis test to determine if the sample mean is significantly different from a given value (usually the population mean, in this case we don't know the population mean). We can use a one-sample t-test since the population standard deviation is unknown. The formula for the t-test statistic is:
t = (x̄ - μ) / (s / √n)
where x̄ is the sample mean, μ is the population mean (unknown), s is the sample standard deviation, and n is the sample size.
Plugging in the values from the given information:
x̄ = 317, s = 37, n = 80, and μ is unknown.
2. The critical value of the test statistic depends on the significance level and the degrees of freedom. Since we don't have the exact population standard deviation, we will use a t-distribution for the critical value. The critical value is found using a t-table or a t-distribution calculator based on the desired significance level (α) and degrees of freedom (df = n - 1).
3. To draw a conclusion about the data, we need to compare the computed test statistic from step 1 with the critical value obtained in step 2. If the computed test statistic falls within the critical region (the region of extreme values determined by the critical value), we reject the null hypothesis. Otherwise, if it falls outside the critical region, we fail to reject the null hypothesis.
4. To calculate the confidence intervals, we can use the t-distribution as well.
For a 95% confidence interval, we can use the formula:
Margin of error = (t * s) / √n,
where t is the critical value for a 95% confidence interval, s is the sample standard deviation, and n is the sample size. The confidence interval is then calculated as:
Confidence interval = (x̄ - margin of error, x̄ + margin of error).
For a 99% confidence interval, we use the same formula but with the critical value for a 99% confidence interval and calculate the margin of error and confidence interval accordingly.
Please provide the desired significance level (α) for the critical value and I can provide the exact calculations for your specific case.