(a) Construct a 90% confidence interval for the population mea...

Question

(a) Construct a 90% confidence interval for the population mean in Exercise 1. Interpret the results. (b) Does it seem likely that the population mean could be within 10% of the sample mean? Explain.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says a survey records the time in minutes past 7 o'clock a.m. that a random sample of 30 students wake up for an early class. The population standard deviation is known to be 25 minutes. Construct a 90% confidence interval for the population mean. Determine if the population mean could reasonably be within 10% of the sample mean, and we're given the data of 60, 75%, 90, 85, 70, 65, 80, 95, 100, 60. 85 90 75 80 70 65 90 1085 80 75 70 65 60 95 100 85 80 75 and 70. So for the first part of this problem, we need to construct a 90% confidence center for the population mean. Now, we were not given the population mean in the problem. But recall that the sample mean is the most unbiased point estimate for the population mean mu. So, what we really need to do is to calculate our sample mean for our data. So recall your formula for the sample mean X bar, that is going to be equal to the sum of X divided by n, where X here is an individual data point and N is our sample size, which in this case is 30. So that means that X bar is going to be equal to 1 divided by 30, multiplied by the quantity, and here we'll sum up all of our values. So we'll have 60 plus 75. Plus 90 + 85 plus 70 plus 65 plus 80. Plus 95. Plus 100 plus 60. Plus 85 plus 90. Plus 75 plus 80. Plus 70. Plus 65 plus 90 plus 100 plus 85 plus 80 plus 75 plus 70 plus 65 plus 60 plus 95 plus 100 plus 85 plus 80. Plus 75 plus 70 in quantity. And when we evaluate this expression, this is going to be approximately equal to 79.2. This has units of minutes since our data was given in minutes. Now we're also going to need to calculate our margin of error in order to construct our confidence interval. So we call your formula for the margin of error, that it's going to be E is equal to Z critical, multiplied by sigma divided by the square root of N. Where E here is our margin of error, Z critical is the critical Z value that corresponds to the 90% confidence interval. Sigma here is a population standard deviation, and in here again is our sample size. Now, note that from the central limit theorem, that when N is greater than or equal to 30, the sampling distribution of sample means can be approximated by a normal distribution. This is why we're using a Z score in this formula. So we just need to look up the critical Z value for a 90% confidence interval, and we see that Z critical is going to be equal to 1.64 5. So, we'll substitute in that value into our formula for the margin of error, and we'll see that E is going to be equal to. 1.645 Multiplied by Sigma, which is a population standard deviation, which we're told in the problem is 25 minutes, so we'll have 25, and this could be divided by the square root of N, which was 30. And when we evaluate this expression, this is going to be approximately equal to 7.5 minutes. So here we just rounded to one decimal place. So, now we have all the information that we need to calculate our 90% confidence interval. So, for our left end point of this interval, we're gonna have X bar minus E. And so this is going to be equal to. 79.2 minus 7.5. Which is going to be equal to. 71.7 minutes. And for our right end point, we're going to have X bar plus E. And this is going to be equal to 79.2 plus 7.5, and this is going to be equal to 86.7 minutes. So we can actually write down our 90% confidence edible, and that is going to be 71.7 minutes is less than you, is less than 86.7 minutes. So this here is going to be the answer for the first part of our problem. So, for the next part, we need to determine if the population mean could reasonably be within 10% of the sample mean. So to be within 10% of the sample mean, that means we need to look at the interval that goes from 0.9, multiplied by X bar to 1.1, multiplied by X bar, and here we've just written that in interval notation. Now, we've already calculated the value for X bar, so we'll just substitute it in, so that means we're going to have the interval that goes from 0.9, multiplied by 79.2. 2 1.1, multiplied by 79.2. And when we do that multiplication, we have the interval that goes from. 71.3. 2 87 0.1. And so here we just round it to one decimal place. And if we compare our two intervals here, we see that our 90% confidence interval falls entirely within the 10% of our mean interval. So that means that it is reasonable to assume that The population mean is within 10% of our sample mean. So that is going to be the answer to the second half of this problem. So just to recap, we found that our 90% confidence interval, or the population mean, is 71.7 minutes is less than mu, which is less than 86.7 minutes. And for the second part of the question, we found that it is reasonable to assume that the population mean is within 10% of the sample mean. So that is going to be the answer for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

(a) Construct a 90% confidence interval for the population mean in Exercise 1. Interpret the results. (b) Does it seem likely that the population mean could be within 10% of the sample mean? Explain.

Key Concepts

Confidence Interval

Population Mean

Margin of Error

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