Question

Body Mass Index. Body mass index (BMI) is a measure of body fat based on height and weight. According to Dietary Guidelines for Americans, published by the U.S. Department of Agriculture and the U.S. Department of Health and Human Services, for adults, a BMI of greater than 25 indicates an above healthy weight (i.e., overweight or obese). The BMIs of 75 randomly selected U.S. adults provided the data on the Weiss Stats site. Use the technology of your choice to do the following.

a. Obtain a normal probability plot, a boxplot, and a histogram of the data.

b. Based on your graphs from part (a). is it reasonable to apply the one-mean z-test to the data? Explain your answer.

c. At the 5% significance level, do the data provide sufficient evidence to conclude that the average U.S. adult has an above healthy weight? Apply the one-mean z-test, assuming a standard deviation of 5.0 for the BMIs of all U.S. adults.

Short answer

Thus, the data provide sufficient evidence to conclude that the average U.S. adult has an above healthy weight.

Step-by-step solution

Step 1. Given

The BMIs of 75 randomly selected U.S. adults provided the data on the Weiss Stats site.

Step 2.  Construct the probability plot

MINITAB procedure:

Step 1: Select Graph > Probability Plot from the drop-down menu.

Step 2: Click OK after selecting Single.

Step 3: In the Graph variables section, input the BMI column.

Step 4: Select OK from the drop-down menu.

Step 2. Minitab output

Step 4. Constructing the boxplot using MINITAB

MINITAB procedure:

Step 1: Select Graph> Boxplot or Stat > EDA > Boxplot from the Graph menu.

Step 2: Select Simple under One Y. Click the OK button.

Step 3: Fill in the BMI data in the graph variables.

Step 4: Select OK from the drop-down menu.

Step 5. MINITAB OUTPUT

Step 6. Construct the histogram by using MINITAB

MINITAB procedure:

Step 1: Choose Graph> Histogram.

Step 2: Choose Simple, and then click OK.

Step 3: In Graph variables, enter the corresponding column of BILL.

Step 4: Click OK.

Step 7. MINITAB output

Step 8. Construct  stem-leaf diagram

MINITAB procedure:

Step 1: Select Graph > Stem and leaf.

Step 2: Select the column of variables in BMI.

Step 3: Select OK.

Step 9. MINITAB OUTPUT

Part (b) Step 10. Conditions to use the z-test are given below

Sample size is small:

When the sample size is fewer than 15, and the variable is normally distributed or extremely close to being normally distributed, the z-test approach is performed.

When the variable is distant from being normally distributed or there is no outlier in the data, the z-test approach is performed when the sample size is between 15 and 30.

When the sample size exceeds 30, the z-test technique is utilized without restriction.

Step 11. Explanation

The sample size is considerable in this case. That means, the number of participants (n) is 75 (>30). Furthermore, the variable's distribution is normally distributed. As a result, using the z-test is reasonable.

Part (c) Step 12. Explanation

1st step:

The null and alternative hypotheses should be stated as follows:

The null hypothesis is:

That is, the data does not support the conclusion that the average adult in the United States is overweight

Hypothesis number two:

H:μ>25

That is, the facts are sufficient to establish that the average adult in the United States is overweight

Step 2: Select a level of relevance.

The significance level is 0.05 in this case.

Step 13. Compute the value of the test statistic 

MINITAB procedure:

Step 1: Select Stat > Basic Statistics > 1-Sample Z from the drop-down menu.

Step 2: In the Samples in Column section, type BMI in the BMI column.

Step 3: Put 5 in the Standard Deviation box.

Step 4: In the Perform hypothesis test box, type 25 for the test mean.

Step 5: Go to Options and change the Confidence Level to 95.

Step 6: As an option, select greater than.

Step 7: In all dialogue boxes, click OK.

Step 14. MINITAB Output

Step 15. Conclusion

5th Step:

Conclusion:

If Psa is true, the null hypothesis must be rejected.

The P-value is 0.044, which is less than the significance level. P(-0.044)a (=0.05) is the result.

As a result, at a 5% level, the null hypothesis is rejected.

As a result, the test results can be concluded to be statistically significant at the 5% level of significance.

6th step:

Interpretation:

As a result, the data give adequate evidence to conclude that the average adult in the United States is overweight.