Question

SAT scores are distributed with a mean of 1,500 and a standard deviation of 300. You are interested in estimating the average SAT score of first year students at your college. If you would like to limit the margin of error of your 95% confidence interval to 25 points, how many students should you sample? Make sure to give a whole number answer. Question Help: Video Message instructor. D Post to forum Submit Question Score: 6/10 answered Question 6 You measure 47 textbooks' weights, and find they have a mean weight of 51 ounces. Assume the population standard deviation is 11.6 ounces. Based on this, construct a 95% confidence interval for the true population mean textbook weight. Give your answers as decimals, to two places Question Help: Message instructor D Post to forum Submit Question Score: 12/20 6/10 answered Question 7 v < If n=26, 7(x-bar)=46, and s=8, find the margin of error at a 95% confidence level Give your answer to two decimal places. Question Help: Message instructor D Post to forum Submit Question Score: 12/20 6/10 answered Question 8 In a survey, 17 people were asked how much they spent on their child's last birthday gift. The results were roughly bell-shaped with a mean of $47 and standard deviation of $18. Construct a confidence Interval at a 99% confidence level Give your answers to one decimal place. + Interpret your confidence interval in the context of this problem. Question Help: Message instructor. Post to forum

Answer 1

To achieve a 95% confidence interval with a margin of error of 25 points, you need to sample approximately 169 students.

For estimating the sample size needed to achieve a certain margin of error in a confidence interval, you can use the formula:

\[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \]

Where:

- \( n \) is the sample size

- \( Z \) is the Z-score corresponding to the desired confidence level (for 95% confidence, \( Z = 1.96 \))

- \( \sigma \) is the population standard deviation (300 in this case)

- \( E \) is the desired margin of error (25)

Plugging in the values:

\[ n = \left( \frac{1.96 \cdot 300}{25} \right)^2 \approx 168.48 \]

Since you can't have a fraction of a student, you would round up to the nearest whole number. Therefore, you should sample around 169 students.

To ensure a 95% confidence interval with a margin of error of 25 points when estimating the average SAT score of first-year students, you need to sample approximately 169 students. This sample size accounts for the population mean of 1,500 and the population standard deviation of 300.