The data set represents the amounts of time (in minutes) spent...

Question

The data set represents the amounts of time (in minutes) spent checking email for a random sample of employees at a company.

c. Repeat part (b), assuming σ = 3.5 minutes. Compare the results.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A gardener tracked the time in units of minutes, spent watering plants for eight randomly selected days. The recorded times were 12.5, 13.2, 14.0, 12.8, 13.6, 11.9, 13.3, 12.6. Assume the population standard deviation is 3.5 minutes. Construct a 90% confidence interval for the meantime spent watering plants per day. Awesome. So it appears for this particular prom, we're ultimately trying to create or construct a 90% confidence interval for the meantime spent watering plants per day based on all the information provided to us by the prom itself. So now that we know that we're ultimately trying to figure out what this 90% confidence interval for the meantime spent watering plants per day is, let's take a moment to read off our multiple choice answers to see what our final answer might be, noting that they're all in the same units of minutes and they're all within parentheses. So they're in interval notation. So A is 10.8.15.1. B is 11.5, 15.7, C is 11.3.14.9, and D is 11.0.15.0. Awesome. So Now that we know what we're trying to solve for, our first step that we need to take is we need to find the sample mean. So we need to determine the value for X bar, and as we should recall, in order to solve for X bar, once again, our sample mean, we need to note that X bar is equal to 12.5 plus 13.2 plus 14.0 plus 12.8 plus 13.6 plus. 11.9 plus 13.3 plus 12.6 all divided by 8 because as we should recall, in order to solve the mean, we take all the values that are given to us in our data set and we divide by the number of values in our data set. So in this case, there's 8 values in our data set, so we add up all 8 of our values and divide by 8. And when we do that, we will find that X bar is equal to 12.9875. And our second step now is we need to Find the Z score for a 90% confidence level. So once again, our second step is that we need to figure out, or rather, we need to find the Z score for a 90% confidence level. And as we should recall, for a 90% confidence interval, the critical Z value. So, you probably could see the critical, once again, our critical Z values usually written as T subscript C. So once again, so using a 90% confidence interval, and when we look it up in our standard normal distribution table, which would be table 4 from your elementary statistics textbook. The 8th edition, we will find that the critical Z value will be equal to 1.645 for a 90% confidence interval. So once again, you need to use a standard normal distribution table in order to figure out what your critical Z value is. So moving right along here. Our next step now, or step 3, is we need to calculate the margin of error. So as we should recall, the margin of error E is equal to Z multiplied by sigma divided by the square root of N, which is equal to when we plug in our known variables, 1.645, which In this case, I said that the critical Z value was TC, but in order to be consistent, we'll call it Z to match our margin of error equation. So once again, our value for Z is 1.645, and we need to multiply it by parenthesis 3.5 divided by the square root of 8. And when we plug that into our calculator, we will find that it's approximately equal to 2.0355. 8 3, so it's gonna be 83. 6 463. Fantastic. So now our fourth step is we need to construct the confidence interval, which, as we should recall, the confidence interval states that X bar plus or minus E, in this case will be equal to 12.9. 875 plus or minus 2.0355 836-463. So that will mean that our confidence interval, which we'll call CI for short, will be approximately equal to parentheses 11.0, 15.0 in parentheses, and this is when we write it in interval notation. And that's it. That is our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be multiple choice answer letter D, which once again is parentheses 11.0. 15.0 minutes because we have to remember that our units are in minutes. I'll make a note of that in our. Answer down below. So make sure when you solve for any problem like this that is when it gives you units, make sure you include the units in your answer. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

The data set represents the amounts of time (in minutes) spent checking email for a random sample of employees at a company.
c. Repeat part (b), assuming σ = 3.5 minutes. Compare the results.

Key Concepts

Standard Deviation (σ)

Sampling Distribution

Confidence Intervals

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