The mean room rate for two adults for a random sample of 26 th...

Question

The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of $31. Assume the population is normally distributed. (Adapted from Expedia)

Construct a 99% confidence interval for the population standard deviation.

Construct a 99% confidence interval for the population standard deviation.

Accepted Answer

Hi, everybody, and glad to have you back. The next question says, a random sample of 18 4-star hotels in a city has a sample standard deviation of $27 for the nightly rate for 2 adults. Assume the population is normally distributed. Construct a 99% confidence interval for the population standard deviation. A 18.45. B 19.44. C. 19.47, or D 16.42. So, we are given, well we have N, our sample, but we are given the standard of sample standard deviation, so S, and we're looking for 99% confidence interval for the population standard deviation or sigma. So, to be able to calculate that, we're going to be using a chi squared. Distribution And when we're estimating the population standard deviation, that formula for the confidence interval. Put in parenthesis, we've got the lower limit as the square root, and underneath the square root sign we have In parentheses N minus 1 multiplied by Squared. Than in the denominator Are chi squared critical value on the right. All under the square root. And then that is going to be the lower limit, so right, lower up here. And then comma and then the upper limit, very similar square root sign. The numerator in parentheses N minus 1 multiplied by Squared, but in the denominator we have that. Left sided critical value pi squared. And close parenthesis. So, obviously, we'll need to calculate these critical values. Why is it that our lower limit, put up by the upper limit, has the right-sided critical value and our upper limit has the left sided? Well, the chi squared critical value is in the denominator. So, chi squared the right side will be the higher number, and so thus, when put in the denominator will make the lower limit. So, let's just make sure we have our distribution in mind, our graph. So, recall with a chi squared distribution, it is not symmetrical, a little bit of chi squared graph there. And so we can't just do a plus or minus uh confidence interval. We need to calculate the lower and upper levels of that confidence interval independently. So we have our chive score distribution, and we'll have a Left sided critical value. Uh lining one tail and then a right sided. Critical value on the right side. With that area of the tail being on the right side of that chi squared value. So we'll go ahead and highlight, so we focus on the fact that we're looking at these outer tails on left and right side. Now, to calculate these critical values, we need our alpha. Well, we have a confidence level rather than an alpha, but that's straightforward enough. We know that alpha. Will equal 1 minus the confidence interval. So our confidence interval is 99%. So Alpha will equal. 0.01. But when we have, as we do here, we have a two-tailed distribution. That means the area. Of each tail is not alpha. But alpha divided by 2. So that's that area on the right, and on the left as well. So now we have what we need to look up that right sided value, that's a little bit simpler. So we would say to get the right sided value, so put R. We have chi squared, and then the alpha that we use on the table would be equal to alpha divided by 2, which is 0.005. So, our chi squad, we're going to look up 0.005, so we put that in the subscript, comma, and now we need our degrees of freedom. So, recall that degrees of freedom, equal N minus 1, and we know we have 18 hotels in the sample, so, N minus 1 equals 17. So back to our chi squared, we have chi squared, and then in the subscript 0.005, 17. So, use those as the alpha and degrees of freedom, look up that value, and we're going to get 35. 0.718 as the right-sided chi squared critical value. Then for the left And chi squared Same degrees of freedom, however, recall that the way the chi square tables oriented it. is giving you information about the area to the right of the critical value. So, not the left tail, but, and we'll draw, I'm going to draw lines. It's going diagonally across that area to the right. Of that left side of critical value, and that includes the right tail, if you look. So we're not looking at the area of the tail on the left, but rather we're looking at the area on the right, so that's striped area. And the area of that. To put area above that I drew box with stripes in it. The area of the striped area is equal to 1 minus alpha divided by 2. So, on the left side, we'll be looking for chi squared, and then in the subscript, this will be 0.995, 1 minus alpha divided by 2. And then still 17 in terms of degrees of freedom. So, when we do that calculation, we get the left sided critical value being 5.69. So, now we need to use our formula for our confidence interval. So, looking at the lower level confidence of the confidence interval, we have the square root sign. Numerator N minus 1 is going to be 17, and it is multiplied by Squad. We're told our sample standard deviation is 27, so we have 27 squared. And in the denominator again. We have the right-handed chi squared critical value. We calculated that as 5. Oops, sorry. See, you gotta be real careful here. The right-sided critical value is 35. 0718. So that amount is going to be less than our sigma, our population standard deviation. Which in turn will be less than, I put the square root sign, under that square root sign, N minus 1, so 17, multiplied by Squared, so 27 squared. All that divided by our denominator the L-sided, the left sided chi squared critical value, which is 5.697. So now we're going to go ahead and plug these values into our calculator. You can pause the video here if you wish to do that on your own. And that's on the right side, going to give us approximately 18.62. Approximately equal to, it must be less than sigma. Which is less than approximately. On the 2nd, the upper limit. That adds up to 46.64. So we can see the answers provided to us in the multiple choice are rounded to the one's place. So, that means our lower limit. Will be approximately 19. That will be less than sigma, which will be less than our upper limit will round up to 47. So that corresponds with choice C. So, once again, we've been provided with information about a sample, it's sample standard deviation. And we're told to assume the population has normal distribution, and then we're supposed to construct a 99% confidence interval for the population standard deviation. So, for sigma. And the end result we have is choice C, 19 and 47. So, our population standard deviation of sigma must be in between 19 and 47. Hope to see you in the next video.

The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of $31. Assume the population is normally distributed. (Adapted from Expedia)

Construct a 99% confidence interval for the population standard deviation.

Key Concepts

Confidence Interval

Sample Standard Deviation

Chi-Square Distribution

Watch next

The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of $31. Assume the population is normally distributed. (Adapted from Expedia)Construct a 99% confidence interval for the population standard deviation.

Key Concepts

Confidence Interval

Sample Standard Deviation

Chi-Square Distribution

Watch next

The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of $31. Assume the population is normally distributed. (Adapted from Expedia)

Construct a 99% confidence interval for the population standard deviation.