The Frayed Nerves Anxiety Scale (FNAS) is used to measure the anxiety level of people on a scale from 0 to 1000. Four students were randomly chosen and given the FNAS before their second exam. Their scores were 700, 653, 740, and 707. The standard deviation sigma of the test is 40.
1. What is x bar?
2. What is the standard deviation of the mean?
3. What is the 95% confidence interval for x bar?
4. What is the 99% confidence interval for x bar?
5. How many students do you need to survey to get the plus or minus for the 95% confidence interval down to 20?
6. How about for the 99% confidence interval?
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1. x = ∑x/n
2. SEm = SD/√n
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3. x ± 1.96 SEm
I'll leave the rest up to you.
To answer these questions, we need to understand some basic concepts of statistics and confidence intervals. Let's go step by step.
1. What is x bar?
- x bar (pronounced as "x bar") represents the sample mean. It is calculated by summing up all the values in the sample and dividing by the number of values. In this case, we have four scores: 700, 653, 740, and 707. Adding them together gives us a total of 2800. Dividing this sum by 4 (the number of scores) gives us x bar = 2800 / 4 = 700. Therefore, x bar is equal to 700.
2. What is the standard deviation of the mean?
- The standard deviation of the mean, denoted as σ(x bar), represents the average amount that the sample mean deviates from the true population mean. To find it, we divide the standard deviation (σ) of the population by the square root of the sample size (n). In this case, σ = 40 (given) and n = 4. Thus, σ(x bar) = σ / √n = 40 / √4 = 40 / 2 = 20. Therefore, the standard deviation of the mean is 20.
3. What is the 95% confidence interval for x bar?
- The 95% confidence interval estimates the range within which the true population mean lies with 95% confidence. To calculate it, we need to determine the margin of error, which is based on the standard deviation of the mean and the desired level of confidence. The formula for the margin of error is (Z * σ(x bar)), where Z is the value from the standard normal distribution corresponding to the desired level of confidence (95% in this case). Looking up this value in a standard normal distribution table, we find that Z = 1.96. Plugging in the values, the margin of error is (1.96 * 20) = 39.2. Next, we subtract the margin of error from x bar to find the lower bound of the confidence interval and add it to x bar to find the upper bound. Therefore, the 95% confidence interval for x bar is (700 - 39.2, 700 + 39.2) or approximately (660.8, 739.2).
4. What is the 99% confidence interval for x bar?
- The process for calculating the 99% confidence interval is identical to the previous step, except that we use a different Z value from the standard normal distribution table. For a 99% confidence level, the corresponding Z value is 2.58. Using this value, the margin of error is (2.58 * 20) = 51.6. Subtracting and adding this margin of error to x bar gives us the 99% confidence interval for x bar, which is approximately (648.4, 751.6).
5. How many students do you need to survey to get the plus or minus for the 95% confidence interval down to 20?
- To determine the required sample size to achieve a given margin of error (or range) for the confidence interval, we can use the formula: n = (Z^2 * σ^2) / (E^2), where Z is the Z-value for the desired confidence level, σ is the standard deviation of the population (given as 40), and E represents the margin of error (given as 20). Plugging in these values, we get n = (1.96^2 * 40^2) / 20^2 = 3.8416 * 1600 / 400 = 15.3664. Since the sample size must be a whole number, we round up to the nearest integer. Therefore, you would need to survey at least 16 students to achieve a plus or minus of 20 for the 95% confidence interval.
6. How about for the 99% confidence interval?
- Using the same formula as in the previous question but with a different Z-value (2.58 for 99% confidence level), we have n = (2.58^2 * 40^2) / 20^2 = 6.6564 * 1600 / 400 = 26.6256. Again, rounding up to the nearest whole number, you would need to survey at least 27 students to achieve a plus or minus of 20 for the 99% confidence interval.