In Exercises 21–24, construct the indicated confidence interva...

Question

In Exercises 21–24, construct the indicated confidence interval for the population mean μ.

c = 0.80, xbar = 20.6, σ = 4.7, n = 100.

Accepted Answer

All right, hello, everyone. So this question says, a sample of 100 cereal boxes has a mean weight of 16.2 ounces. If the population standard deviation is 3.5 ounces, construct the 80% confidence level for the population mean weight of the cereal boxes. And here we've got 4 different answer choices labeled A through D. So first, what do we know? Well, our sample mean, first of all, or X bar is equal to 16.2 ounces. Our population standard deviation, sigma is equal to 3.5 ounces. Our sample size or N is equal to 100. And we already know that our confidence level is 80%. So now, our critical Z value is Z of alpha divided by 2. Recall that alpha is equal to one subtracted by the confidence level expressed as a decimal. So that's One subtracted by 0.80, which gives you 0.20. 0.20 divided by 2. Gives you Z of 0.10. So according to our standard normal table, our critical value is equal to 1.28. And now you can calculate the standard error. So the standard error or SE is equal to sigma divided by the square root of N. So that's 3.5 divided by the square root of 100. And that's equal to 0.35. Now to construct the confidence interval. So the confidence interval Is equal to X bar. Added to and subtracted by. Z of alpha divided by 2. Multiplied by the standard error. So this gives you 16.2. Added to and subtracted by. 1.28. Multiplied by 0.35. Simplifying this expression further. Equals 16.2. Added to and subtracted by 0.448. So from this, you get your two numbers or the two boundaries of the confidence interval. The lower one from subtracting 0.448, and the higher one from adding 0.448. Ultimately, your confidence level comes out to 15.8 as the lower bound. And 16.6 as the upper bound after rounding to one decimal place. This corresponds to option A, the multiple choice, making that your correct answer. And there you have it. So with that being said, thank you so very much for watching. And I hope you found this helpful.

In Exercises 21–24, construct the indicated confidence interval for the population mean μ.
c = 0.80, xbar = 20.6, σ = 4.7, n = 100.

Key Concepts

Confidence Interval

Standard Deviation

Sample Size

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