Select a random sample of 30 student responses to question 6, "How many credit hours are you taking this term?" Using the information from this sample, and assuming that our data set is a random sample of all Kaplan statistics students, estimate the average number of credit hours that all Kaplan statistics students are taking this term using a 95% level of confidence. Be sure to show the data from your sample and the data to support your estimate.
To estimate the average number of credit hours that all Kaplan statistics students are taking this term with a 95% level of confidence, we need to perform the following steps:
1. Randomly select a sample of 30 student responses to question 6, "How many credit hours are you taking this term?" from the data set.
2. Calculate the mean, standard deviation, and sample size (n) from the sample data.
3. Determine the appropriate critical value for a 95% confidence level. Since the sample size (n = 30) is relatively small and the population standard deviation is unknown, we will use a t-distribution. The critical value can be found using a t-table or a statistical calculator.
4. Calculate the margin of error by multiplying the critical value by the standard deviation divided by the square root of the sample size.
5. Compute the lower and upper bounds of the confidence interval by subtracting and adding the margin of error to the sample mean.
6. Finally, present the sample data, sample mean, margin of error, and the confidence interval as the estimate of the average number of credit hours for all Kaplan statistics students.
Here's an example of how to perform these steps:
1. Randomly select a sample of 30 student responses to question 6, "How many credit hours are you taking this term?" from the data set:
- Sample data: 15, 12, 16, 14, 17, 13, 14, 15, 16, 13, 12, 15, 13, 14, 16, 13, 14, 15, 16, 13, 12, 15, 14, 13, 16, 15, 18, 14, 12, 15.
2. Calculate the mean, standard deviation, and sample size (n) from the sample data:
- Mean (x̄): Calculate the sum of all the observations and divide it by the sample size (n).
- Standard deviation (s): Calculate the square root of the sum of the squared differences between each observation and the mean, divided by (n-1).
- Sample size (n): The number of observations in the sample.
3. Determine the appropriate critical value for a 95% confidence level:
- Since the sample size (n = 30) is relatively small and the population standard deviation is unknown, we will use a t-distribution. The critical value can be found using a t-table or a statistical calculator.
- For a 95% confidence level with 29 degrees of freedom (n-1), the critical value is approximately 2.045.
4. Calculate the margin of error by multiplying the critical value by the standard deviation divided by the square root of the sample size:
- Margin of error = (critical value) * (standard deviation / √n).
5. Compute the lower and upper bounds of the confidence interval:
- Lower bound = sample mean - margin of error.
- Upper bound = sample mean + margin of error.
6. Finally, present the sample data, sample mean, margin of error, and the confidence interval as the estimate of the average number of credit hours for all Kaplan statistics students:
- Sample mean (x̄): Calculate the mean credit hours from the sample data.
- Margin of error: Calculate the margin of error using the critical value, standard deviation, and sample size.
- Confidence interval: The range of values within which the true population mean (average number of credit hours) is likely to fall with 95% confidence.
For example, let's assume that the sample mean (x̄) is calculated to be 14.5 and the standard deviation (s) is 1.5. Also, using the t-distribution table with 29 degrees of freedom, the critical value is 2.045.
Using these values, we can calculate the margin of error as:
- Margin of error = (2.045) * (1.5 / √30) ≈ 0.565.
The confidence interval would then be:
- Lower bound = 14.5 - 0.565 ≈ 13.935.
- Upper bound = 14.5 + 0.565 ≈ 15.065.
Therefore, with 95% confidence, we can estimate that the average number of credit hours all Kaplan statistics students are taking this term is between approximately 13.935 and 15.065 credit hours.