91Ó°ÊÓ

As part of the National Health and Nutrition Examination Survey, the Department of Health and Human Services obtained self-reported heights (in.) and measured heights (in.) for males aged 12-16. Listed below are sample results. Construct a \(99 \%\) confidence interval estimate of the mean difference between reported heights and measured heights. Interpret the resulting confidence interval, and comment on the implications of whether the confidence interval limits contain \(0 .\) $$ \begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { Reported } & 68 & 71 & 63 & 70 & 71 & 60 & 65 & 64 & 54 & 63 & 66 & 72 \\ \hline \text { Measured } & 67.9 & 69.9 & 64.9 & 68.3 & 70.3 & 60.6 & 64.5 & 67.0 & 55.6 & 74.2 & 65.0 & 70.8 \\ \hline \end{array} $$

Step by step solution

01

- Calculate the differences

Find the difference between the reported and measured heights for each individual. This can be done by subtracting the measured heights from the reported heights.\( \text{Differences} = [68 - 67.9, 71 - 69.9, 63 - 64.9, 70 - 68.3, 71 - 70.3, 60 - 60.6, 65 - 64.5, 64 - 67, 54 - 55.6, 63 - 74.2, 66 - 65, 72 - 70.8] \)

02

- Calculate the mean of the differences

Find the sample mean of these differences. First, sum up all the differences and then divide by the number of differences (12).\[ \text{Mean difference} = \frac{\text{Sum of all differences}}{12} \]

03

- Calculate the standard deviation of the differences

Calculate the sample standard deviation. This can be done using the formula: \[ s = \frac{1}{n-1} \times \text{sum of squared deviations from the mean} \]

04

- Determine the standard error of the mean difference

The standard error of the mean difference (SE) is given by: \[ SE = \frac{s}{\root n} \] where \( s \) is the standard deviation and \( n \) is the number of samples (12).

05

- Find the critical value for a 99% confidence interval

For a 99% confidence interval and a sample size of 12, the critical value (\( t \)) can be found using a t-distribution table. For \( n - 1 \) degrees of freedom (\( 11 \)), the critical \( t \)-value is approximately 3.106.

06

- Calculate the margin of error

The margin of error (ME) can be calculated as: \[ ME = t \times SE \]

07

- Construct the confidence interval

Finally, construct the confidence interval using the mean difference, margin of error, and the formula: \[ \text{Confidence Interval} = \bar{x} \text{difference} \times ME \]

08

- Interpret the confidence interval

Interpret the meaning of the confidence interval. Determine whether 0 is within the interval and discuss the statistical significance.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Difference

To start, we need to understand what the mean difference is. When comparing two sets of data, such as reported heights vs. measured heights, we calculate the difference for each pair. For example, if the reported height is 68 inches and the measured height is 67.9 inches, the difference is 0.1 inches. We do this for all pairs. Next, we find the average of these differences. This average is the mean difference. It helps us understand the general discrepancy between the reported and measured data. By summing all individual differences and dividing by the number of differences, we obtain the mean difference.

Standard Deviation

Once we have the differences, the next step is to determine their consistency or spread. The standard deviation tells us how much the values deviate from the mean difference. In other words, it shows the variability within our set of differences. To find the standard deviation, we subtract each difference from the mean difference, square the result, sum all squared results, divide by the number of pairs minus one, and finally take the square root of the sum. This process quantifies the dispersion, helping us understand the extent of variation in our differences.

T-Distribution

When constructing a confidence interval, especially with smaller sample sizes, we use the t-distribution. Unlike the normal distribution, the t-distribution accounts for small sample sizes, making it more accurate under these conditions. For our case with 12 samples, we use 11 degrees of freedom (sample size minus one). The t-distribution gives us a critical value that helps in calculating the margin of error. This critical value depends on the confidence level we wish to achieve. For a 99% confidence level, this value is approximately 3.106.

Margin of Error

The margin of error reflects the precision of our estimate. It is directly related to the t-distribution critical value and the standard error of the mean difference. To compute the margin of error, multiply the t-distribution critical value by the standard error. The standard error is calculated by dividing the standard deviation by the square root of the number of samples. So, the margin of error tells us how far our sample mean difference might be from the true population mean difference with a 99% confidence level.

Statistical Significance

Statistical significance helps us determine the importance of our results. After constructing the confidence interval using the mean difference and the margin of error, we examine whether this interval includes 0. If 0 is not included, it suggests a significant difference between reported and measured heights. In our example, if the confidence interval ranges from 0.5 to 2 inches, it implies a meaningful discrepancy between the two sets of data. Such findings are statistically significant, indicating that the observed differences are not due to random chance.