On a particular day a consumer-advice bureau received 125...

On a particular day a consumer-advice bureau received 125 calls. For a random sample of 40 of these calls, it was found that mean time taken in providing the requested advice was 7.28 minutes, and the sample standard deviation was 5.32 minutes. a. Find a $99 \%$ confidence interval for the mean time taken per call. b. Find a $90 \%$ confidence interval for the total amount of time taken in answering these 125 calls.

On a particular day a consumer-advice bureau received 125 calls. For a random sample of 40 of these calls, it was found that mean time taken in providing the requested advice was 7.28 minutes, and the sample standard deviation was 5.32 minutes.
a. Find a $99 \%$ confidence interval for the mean time taken per call.
b. Find a $90 \%$ confidence interval for the total amount of time taken in answering these 125 calls.

Key Concepts

Linear Transformation of Confidence Intervals

When a parameter of interest is a linear function of another parameter (for example, a total that is a constant multiple of a mean), the confidence interval for the transformed parameter can be obtained by applying the same linear transformation to the original interval. This property allows confidence intervals for means to be scaled to provide intervals for totals or other related quantities.

Standard Error

The standard error is the estimated standard deviation of the sampling distribution of a statistic (commonly the sample mean). It measures how much the sample mean is expected to vary from the true population mean. A smaller standard error indicates more precise estimates, and it is influenced by both the sample variability and the sample size.

t-Distribution

The t-distribution is used for estimating population parameters when the sample size is relatively small or when the population standard deviation is unknown. It accounts for additional uncertainty compared to the normal distribution, particularly through its degrees of freedom, which are related to the sample size. This distribution is critical in constructing confidence intervals when dealing with sample means.

Confidence Interval

A confidence interval is a range derived from sample data that is used to estimate a population parameter. It provides an upper and lower bound which, with a specified level of confidence (e.g., 99% or 90%), is believed to contain the true parameter. The interval is constructed using the sample statistic (such as a mean), the variability in the data, and a critical value from a statistical distribution.

Margin of Error

The margin of error reflects the extent of uncertainty around a sample estimate, providing the range by which the observed statistic may differ from the true population parameter. It is calculated by multiplying a critical value (from the t- or normal distribution) by the standard error. This quantity helps determine the width of the confidence interval.

Transcript

00:01 So we have a scenario in which we have a population size of 125 of which they've taken a random sample of 40 and we know that the mean of those 40 is 7.28 minutes and the sample standard deviation is 5.32 minutes.

00:22 And i'm gonna double check on that number.

00:24 That seems pretty big, but it is what it is.

00:27 That is correct.

00:28 And in part a we want to find a 99% confidence interval for the mean.

00:37 And so because we have more than our we take five% 7.28 plus or minus.

00:55 And the z star value that we use for 99% confidence is that 2.576.

01:03 Next we need to multiply by that standard air multiplied by that correction factor and that finite population correction factor.

01:13 So we're going to have that 5.32 divided by the square root of n, which was 40 and then times the square root of the 125 minus the sample size over one less than the sample size.

01:30 Excuse me.

01:30 The population size.

01:32 So when we do that calculation we have 7.28 -2.576 times the 5.32 divided by the square root of 40.

01:43 And then we need to multiply that by that correction factor and we have the numerator will be 85 underneath the radical divided by 1 24.

01:54 And when we get that calculation, the lower limit comes out to be five 4, 8 6 minutes and up to and will change that subtraction sign to an addition sign and that addition sign yields 9.074 minutes.

02:16 So we are 99% confident that that means of the populations lies somewhere in there.

02:22 Now on part b we want to look at 90% confidence interval for the total number of minutes.

02:31 That total number of minutes of, of minutes.

02:34 And and so we're going to take that total number of minutes which is going to be that 125 times our sample mean and our sample mean was at 7.28.

02:49 And we can get what that total is 125 times...

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