Question

Table $11.42$contains information from a survey among $499$participants classified according to their age groups. The second column shows the percentage of obese people per age class among the study participants. The last column comes from a different study at the national level that shows the corresponding percentages of obese people in the same age classes in the USA. Perform a hypothesis test at the $5$% significance level to determine whether the survey participants are a representative sample of the USA obese population.

Age Class (Years)	Obese Expected (percentage)	Obese Observed (Frequencies)
$20-30$	$22.4$	$122$
$31-40$	$18.6$	$104$
$41-50$	$12.8$	$78$
$51-60$	$10.4$	$64$
$61-70$	$35.8$	$168$

Table$11.42$

Short answer

Decision: Reject the null hypothesis

Reason for decision: Because P-value$<\alpha$

Conclusion: There is adequate evidence to confirm that the distribution of the survey participants is not a representative sample of the USA obese population.

Step-by-step solution

Given Information

Perform a hypothesis test at the $5$% significance level to determine whether the survey participants are a representative sample of the USA obese population.

Apply null hypothesis and alternative hypothesis

The null hypothesis is stated below:

H_$0$ : The survey participants are a representative sample of the USA obese population.

Against the alternative hypothesis as shown below:

H_$\alpha$ : The survey participants are not a representative sample of the USA obese population.

The goodness-of-fit test can be completed here because all entries are greater than or equal to five. Therefore, the degree of freedom can be computed as shown below:

d f= number of cells -$1$

$=5-1$

Test Statistics

From the above calculation, it is clear that the distribution for the test is$x_4^2$

Do the calculation in excel as follow:

Hence, the test statistic is 85.45

Find the  P value

The p-value can be calculated in excel by using the CHIDIST ( ) formula as shown below:

Hence, the P-value is $0.000$

Graphical Representation

The chi-square sketch is shown below