Question

in a recent year, a health agency reported that 35.8% of adults in a certain large county are obese. a county health service planning a new awareness campaign polls a random sample of 400 adults living there. in this sample, 128 people were found to be obese based on their answers to a health questionnaire. do these responses provide strong evidence that the 35.8% figure is not accurate for this region? correct the mistakes you find in the accompanying student’s attempt to test an appropriate hypothesis. consider an event to be rare if its probability of occurring is less than 0.01. click the icon to view the student’s attempt to test an appropriate hypothesis. c. this condition is satisfied because the sample was a simple random sample. d. this condition is satisfied because n = > 10% of the county population. e. this condition is satisfied because np = ≤10 and nq = ≤10. f. this condition is not satisfied. the observed proportion of successes in the sample is \\(\frac{128}{400}\\) = 0.320. (round to three decimal places as needed.) show calculations for the appropriate measure of spread for the sampling distribution for this test. = (round to three decimal places as needed.)

Explanation:

Step1: Check 10% population condition

We verify if the sample size $n=400$ is ≤10% of the large country's adult population. Since the population is large, $400 \leq 0.1N$ (where $N$ is the large national adult population) holds true.

Step2: Check np/nq ≥10 condition

Calculate $np$ and $nq$ using $p=0.358$, $q=1-p=0.642$, $n=400$:
$np = 400 \times 0.358 = 143.2$
$nq = 400 \times 0.642 = 256.8$
Both values are >10, so this condition is satisfied.

Step3: Calculate standard error

Use the formula for the standard deviation of the sampling distribution: $\sqrt{\frac{p_0(1-p_0)}{n}}$, where $p_0=0.358$, $n=400$.
$\sqrt{\frac{0.358 \times 0.642}{400}}$
First compute numerator: $0.358 \times 0.642 = 0.229836$
Then divide by $n$: $\frac{0.229836}{400} = 0.00057459$
Take square root: $\sqrt{0.00057459} \approx 0.024$

Step4: Identify correct condition option

The correct condition satisfaction is that the 10% population condition is met (option D), as the sample is ≤10% of the large population.

Answer:

Correct Condition Option:

D. This condition is satisfied because n = 400 > 10% of the county population.

Standard Error:

0.024