In Exercises 9–12, construct the indicated confidence interval...

Question

In Exercises 9–12, construct the indicated confidence interval for the population mean μ using the t-distribution. Assume the population is normally distributed.

c = 0.99, xbar = 24.7, s = 4.6, n = 50

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says a nutritionist analyzes the protein content in grams of a sample of 7 energy bars and finds a sample mean of 9.8 g with a sample standard deviation of 2.7 g. Assuming the protein content is normally distributed, construct a 90% confidence interval for the population mean using the T distribution. So, the first thing we want to do is identify the information that we're given in this problem. And the first piece of information that we're given is our sample size, which we'll label with N. And our sample size here, in this case is going to be equal to 7, so that's we're sampling 7 energy bars. The next piece of information that we're given is our sample mean, which we'll label as X bar, and this is going to be equal to 9.8 g. And the next piece of information is our sample standard deviation, which we'll label as S, and this is going to be equal to 2.7 g. So we're asked in this problem to construct a confidence interval, and for the confidence interval, we're gonna need our margin of error, and for a margin of error, we're gonna need to first calculate our standard error of the mean. And recall your formula for the standard error, that is going to be our standard error SE is going to be equal to S divided by the square root of N. And when we substitute in our values, we see that our standard error is going to be equal to 2.7 divided by the square root of 7. So we'll now use this to calculate our margin of error, and recall your formula for the margin of error, that is going to be E is equal to. T, multiplied by our standard error, SE or T here is our critical T value, since we're using a T distribution. Now, our critical T value is something that we need to look up in our tables, and in order to do that, we first need to determine the degrees of freedom. And so recall that for the degrees of freedom, which we'll label as DF, that that is going to be equal to N minus 1, which in our case is going to be equal to 7 minus 1, which is equal to 6. So, when we look up on our tables, or our critical T value, T star. For the 90% confidence level, with 6 degrees of freedom, we see the T is going to be equal to 1.943. So we'll now substitute that value into our formula for the margin of error. That means that E is going to be equal to. 1.9 or 3, multiplied by 2.7 divided by the square root of 7. And when we evaluate this expression, this is going to be approximately equal to 2.0, and here we just round it to one decimal place. So, now we can use our margin of error to write our confidence interval. So, for the 90% confidence interval, our confidence interval is going to be written as X bar, plus or minus E, and substituting in our values. This is going to be equal to. 9.8 Grams plus or minus. 2.0 g. And we actually want to write this in interval notation. So, we're going to have the interval, going from 9.8 g minus 2.0 g. 2 9.8 g. Plus 2.0 g. And evaluating that expression, this is going to be equal to the interval, from 7.8 g to 11.8 g. So this here is the answer that we were looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

In Exercises 9–12, construct the indicated confidence interval for the population mean μ using the t-distribution. Assume the population is normally distributed.
c = 0.99, xbar = 24.7, s = 4.6, n = 50

Key Concepts

Confidence Interval

t-Distribution

Sample Mean and Standard Deviation

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