After studying and reviewing the concepts and practices of Module 3, answer the following questions:
1) A sample size of 35 scores has a mean score of 85 points with a standard deviation of 3.25 points.
A. Find a 90% confidence interval for the scores population mean and explain your results.
B. Find a 95% confidence interval for the scores population mean and explain your results.
C. What would be the sample size if an error of 5% is desired?
2) A random sample of 9 young adults was selected and the weight height was calculated to be 65.7 inches. Since there is no information for the population standard deviation, the sample standard deviation was also calculated, and the value was 3.46 inches. If the population grades follow a normal distribution.
A. Find a 95% confidence interval for the height population mean and explain your results.
B. Find a 99% confidence interval for the height population mean and explain your results.
Full Answer Section
Therefore, the 90% confidence interval for the scores population mean is:
85 ± 1.645 * (3.25 / √35)
This means that we are 90% confident that the population mean score is between 82.606 and 87.394.
B. Find a 95% confidence interval for the scores population mean and explain your results.
To find a 95% confidence interval for the scores population mean, we can use the same formula as above, but with a different critical value.
For a 95% confidence interval, the critical value is 1.96.
Therefore, the 95% confidence interval for the scores population mean is:
This means that we are 95% confident that the population mean score is between 82.250 and 87.750.
C. What would be the sample size if an error of 5% is desired?
To calculate the sample size needed to achieve a desired error margin, we can use the following formula:
(critical value)^2 * (standard deviation)^2 / (error margin)^2
where critical value is the critical value for the desired confidence interval, standard deviation is the population standard deviation, and error margin is the desired error margin.
Since we do not know the population standard deviation, we can use the sample standard deviation as an estimate.
Assuming we want a 90% confidence interval and an error margin of 5%, the required sample size is:
(1.645)^2 * (3.25)^2 / (0.05)^2
Therefore, we would need a sample size of at least 784 to achieve a 90% confidence interval with an error margin of 5%.
2) A random sample of 9 young adults was selected and the weight height was calculated to be 65.7 inches. Since there is no information for the population standard deviation, the sample standard deviation was also calculated, and the value was 3.46 inches. If the population grades follow a normal distribution.
Find a 95% confidence interval for the weight height population mean.
To find a 95% confidence interval for the weight height population mean, we can use the following formula:
mean ± t * (standard deviation / √sample size)
where t is the critical value for a 95% confidence interval and a sample size of 9. This value can be found using a t-table.
For a 95% confidence interval and a sample size of 9, the critical value is 2.262.
Therefore, the 95% confidence interval for the weight height population mean is:
65.7 ± 2.262 * (3.46 / √9)
This means that we are 95% confident that the population weight height mean is between 62.550 and 68.850 inches.
Conclusion
We can use confidence intervals to estimate the population mean based on a sample of data. The confidence interval provides a range of values that are likely to contain the population mean. The wider the confidence interval, the less precise the estimate.
The sample size needed to achieve a desired error margin is determined by the critical value for the desired confidence interval, the population standard deviation, and the desired error margin.