The data set represents the scores of 12 randomly selected stu...

Question

The data set represents the scores of 12 randomly selected students on the SAT Physics Subject Test. Assume the population test scores are normally distributed and the population standard deviation is 108. (Adapted from The College Board)

b. Construct a 90% confidence interval for the population mean. Interpret the results.

Data set of 12 SAT Physics Subject Test scores: 590, 650, 730, 560, 460, 400, 620, 780, 510, 700, 590, 670.

b. Construct a 90% confidence interval for the population mean. Interpret the results.

Accepted Answer

Hello, everyone, let's take a look at this question together. A training coordinator collected the math test scores out of 800 of 12 randomly selected applicants for a tech internship program. The test scores are assumed to follow a normal distribution, and the population standard deviation is known to be 108. The scores are 672, 701, 715, 694, 689, 728, 645, 710, 668, 675, 703, and 691. Using a 90% confidence level, struct a confidence interval to estimate the population mean score, then interpret the result. So in order to solve this question, we have to recall how to construct a confidence interval so that we can construct one to estimate the population mean score of our scores that we are given for the 12 randomly selected applicants out of a total of 800, where we know that the test scores are assumed to follow a normal distribution. The population standard deviation is known to be 108, and we are using a 90% confidence level. And so in order to solve this question and construct our confidence interval, we know that we are going to use the Z interval formula since the population standard deviation is known and the sample size is small, but the district. is normal. That is why we use the Z interval formula, where we have the following formula, where we know that the formula includes the sample mean, the Z critical value for the given confidence level, the population standard deviation, and the sample size. And so the first step is to calculate the sample mean. which we know that the sample mean is calculated by adding all of the test scores divided by the number of test scores, which is equal to 12, so our sample mean is approximately 690.92. And then we need to identify the Z value. For 90% confidence. So with a confidence level of 90%, our Z score for an alpha divided by 2, which is equal to our Z for 0.05, using the standard normal distribution table, our Z score is 1.645. And then the next step is to calculate the margin of error. Which is given as E is equal to our Z score multiplied by the population standard deviation, divided by the square root of our sample size. Which substituting the values into the formula for the margin of error, our value for the margin of error is approximately 51.29. And now we can construct our confidence interval, which is given as the confidence interval is equal to the sample mean plus or minus the margin of error. So we have 690.92 plus minus 51.29. So our confidence interval is. 639.6 to 742.2. And our interpretation of this result is that we are 90% confident that the true average math test score for all internship applicants lies between 639.6 and 742.2. And looking at our answer choices, we can identify that answer choice B is the correct answer. As answer choice B says, 639.6 to 742.2, and this interval likely contains the population mean. So answer choice B is the correct answer, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Confidence Interval

Normal Distribution

Standard Deviation

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