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 = 0.95, xbar = 31.39, σ = 0.80, n = 82.

Accepted Answer

All right. Hello, everyone. So this question says, a sample of 60 cartons of eggs has a mean weight of 24.5 ounces. The population standard deviation is 1.2 ounces. Construct the 95% confidence interval for the population mean weight of the cartons. And here we have 4 different answer choices labeled A through D. So first, let's take a moment to find what we know. First, we know that X bar, or sample mean, is 24.5 ounces. And our population standard deviation, that's sigma. Is equal to 1.2 ounces. Our sample size n is equal to 60. And we already know that the confidence level is 95%. So first, we have to find the critical Z value. So the Z value would be Z of alpha divided by 2. Where Alpha is equal to one subtracted by the confidence level as a decimal. So For a confidence level of 95%, alpha is equal to. 0.05. So alpha divided by 2 is 0.025. And so our critical value, according to the standard normal distribution table. Is 1.96. So now we can calculate the standard error. Standard error or SE is equal to sigma divided by the square root of N. So that's equal to 1.2, divided by the square root of 60. Which equals approximately 0.155. Now for the confidence interval. The confidence interval, or CI is equal to X bar, added to and subtracted by. Z of alpha divided by 2. Multiplied by the standard error. So that's equal to 24.5, added to and subtracted by. 1.96 multiplied by 0.155. Simplifying this expression further, you get 24.5. Added to and subtracted by. 0.304. And from here you get two numbers, that is the two boundaries of the confidence interval, the lower one from subtracting 0.304, and the upper one from adding 0.304. So ultimately, after rounding to one decimal place, your confidence interval should come out to 24.2, being the lower bound. And 24.8 being the upper bound. This corresponds to option D in the multiple choice, making that the 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 = 0.95, xbar = 31.39, σ = 0.80, n = 82.

Key Concepts

Confidence Interval

Sample Mean (x̄)

Standard Deviation (σ) and Sample Size (n)

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