Question

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% of adults in this large county are obese? a student’s attempt to test an appropriate hypothesis is shown below. correct the mistakes you find in the accompanying student’s attempt. 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. check the 10% condition. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. this condition is satisfied because np = \\(\square\\) ≥ 10 and nq = \\(\square\\) ≥ 10. b. this condition is satisfied because np = \\(\square\\) ≤ 10 and nq = \\(\square\\) ≤ 10. c. this condition is satisfied because the sample was a simple random sample. d. this condition is satisfied because n = \\(\square\\) > 10% of the county population. e. this condition is satisfied because n = \\(\square\\) < 10% of the county population. f. this condition is not satisfied.

Explanation:

Step1: Define 10% Condition rule

The 10% condition requires the sample size $n$ to be less than or equal to 10% of the total population to ensure independence of samples.

Step2: Identify sample size

The given sample size is $n=400$. For the 10% condition to be satisfied, $400$ must be $<10\%$ of the county population.

Step3: Evaluate options

Option E states the condition is satisfied because $n=400 < 10\%$ of the county population, which aligns with the correct 10% condition. Option D incorrectly uses $>$, so it is wrong.

Answer:

E. This condition is satisfied because n = 400 < 10% of the county population.