The mean number of hours of flying time for pilots at...

The mean number of hours of flying time for pilots at Continental Airlines is 49 hours per month (The Wall Street Jourmal, February $25,2003$ ). Assume that this mean was based on actual flying times for a sample of 100 Continental pilots and that the sample standard deviation was 8.5 hours. $$ \begin{array}{l}{\text { a. } \text { At } 95 \% \text { confidence, what is the margin of error? }} \\ {\text { b. What is the } 95 \% \text { confidence interval estimate of the population mean flying time for }} \\ {\text { the pilots? }}\end{array} $$ $$ \begin{array}{l}{\text { c. The mean number of hours of flying time for pilots at United Airlines is } 36 \text { hours per }} \\ {\text { month. Use your results from part (b) to discuss differences between the flying times }} \\ {\text { for the pilots at the two airlines. The Wall Street Journal reported Anited Airlines as }}\end{array} $$ $$ \begin{array}{l}{\text { having the highest labor cost among all airlines. Does the information in this exercise }} \\ {\text { provide insight as to why United Airlines might expect higher labor costs? }}\end{array} $$

The mean number of hours of flying time for pilots at Continental Airlines is 49 hours per
month (The Wall Street Jourmal, February $25,2003$ ). Assume that this mean was based on
actual flying times for a sample of 100 Continental pilots and that the sample standard
deviation was 8.5 hours.
$$
\begin{array}{l}{\text { a. } \text { At } 95 \% \text { confidence, what is the margin of error? }} \\ {\text { b. What is the } 95 \% \text { confidence interval estimate of the population mean flying time for }} \\ {\text { the pilots? }}\end{array}
$$
$$
\begin{array}{l}{\text { c. The mean number of hours of flying time for pilots at United Airlines is } 36 \text { hours per }} \\ {\text { month. Use your results from part (b) to discuss differences between the flying times }} \\ {\text { for the pilots at the two airlines. The Wall Street Journal reported Anited Airlines as }}\end{array}
$$ $$
\begin{array}{l}{\text { having the highest labor cost among all airlines. Does the information in this exercise }} \\ {\text { provide insight as to why United Airlines might expect higher labor costs? }}\end{array}
$$

Key Concepts

Margin of Error

This is the amount added and subtracted from the sample mean to create a confidence interval for a population parameter. It is determined by the critical value from the sampling distribution (often from the z- or t-distribution), multiplied by the standard error of the estimate. The margin of error provides a range within which the true population parameter is expected to lie with a specified level of confidence.

Confidence Interval

A confidence interval is a range of values used to estimate an unknown population parameter. It is calculated by taking the sample mean and adding and subtracting the margin of error. The interval reflects the uncertainty inherent in using a sample to estimate a population value, and the chosen confidence level (such as 95%) indicates the proportion of such intervals that would contain the true parameter if repeated samples were taken.

Standard Error

The standard error of the mean measures the dispersion of sample means around the population mean, quantifying the precision of the sample mean as an estimate. It is calculated by dividing the sample standard deviation by the square root of the sample size. A smaller standard error indicates that the sample mean is likely to be a more accurate reflection of the population mean.

Central Limit Theorem

The Central Limit Theorem states that the sampling distribution of the sample mean will approach a normal distribution as the sample size becomes large, regardless of the original population distribution. This theorem underpins the use of normal-based confidence intervals and margin of error calculations in estimating population parameters from sample statistics.

Comparative Statistical Inference

Comparative statistical inference involves comparing the parameter estimates (such as means) from different populations to draw conclusions about differences between them. In the context of confidence intervals, this might involve examining whether the intervals from separate samples overlap or whether one mean lies outside the confidence interval of another, thereby suggesting a statistically significant difference between the groups.

Transcript

00:01 All right, this question gives us some information about continental airlines, working hours, and the first part wants us to find the margin of error for a 95 % confidence interval.

00:19 So remember that error equals plus or minus a special t star value.

00:30 We're dealing with t -star because we don't have access to the population standard deviation times the standard error.

00:42 So, the only thing we don't know right now is t -star.

00:47 So to find that, we can either use a table or drawing a normal curve.

00:56 For a 95 % interval, we know that the middle of the curve, needs to contain 0 .95 area.

01:04 So that means that there's 0 .025 in each tail.

01:16 So from there, we know that our t star is inverse t of, we want to an area to the left of 0 .975 and how many degrees of freedom we have is just one minus the sample size or one less than the sample size sorry which in this case our t star value is 1 .984 all right so now we can plug in to our formula so we know that error equals plus or minus our t star value that we found earlier 1 .984 times times our sample standard deviation over the square root of our sample size, which is, in this case, in either direction.

03:11 So now it wants us to generate the interval.

03:18 So part b wants the 95 % interval.

03:27 And this is pretty easy to do now because we know that our interval is just x bar plus or a interval.

03:37 Minus our margin of error, which we already calculated.

03:46 So that's 49 plus or minus 1 .684, which gives us an interval from 40 7 .3136 to 50 .68.

04:21 So that's our 95 % confidence interval for the population mean...

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