Question

the graph shows the results of a survey of adults in country a, ages 33 to 51, who were asked if they participated in a sport. seventy percent of adults said they regularly participated in at least one sport, and they gave their favorite sport. you randomly select 300 people in country a, ages 33 to 51, and ask them if they regularly participate in at least one sport. you find that 60% say no. how likely is the result? do you think this sample is a good one? explain your reasoning. select the correct choice below and fill in the answer box within your choice. (round to four decimal places as needed.) a. the result is highly likely because its probability, , is less than 0.05. thus, the sample is not a good representative sample. b. the result is highly unlikely because its probability, , is less than 0.05. thus, the sample is not a good representative sample. c. the result is highly unlikely because its probability, , is greater than 0.05. thus, the sample is a good representative sample. d. the result is highly likely because its probability, , is greater than 0.05. thus, the sample is not a good representative sample.

Explanation:

Step1: Calculate the mean and standard deviation

Given \(p = 0.7\) (proportion of adults who participate in at least one sport), \(n=300\).
The mean of the sampling distribution of the sample proportion \(\hat{p}\) is \(\mu_{\hat{p}}=p = 0.7\).
The standard deviation is \(\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}=\sqrt{\frac{0.7\times(1 - 0.7)}{300}}=\sqrt{\frac{0.7\times0.3}{300}}=\sqrt{\frac{0.21}{300}}\approx0.0265\).

Step2: Calculate the z - score

The sample proportion \(\hat{p}=1 - 0.6=0.4\) (since 60% say no, so 40% say yes).
The z - score is \(z=\frac{\hat{p}-\mu_{\hat{p}}}{\sigma_{\hat{p}}}=\frac{0.4 - 0.7}{0.0265}=\frac{- 0.3}{0.0265}\approx - 11.32\).
Using the standard normal distribution table, \(P(\hat{p}\leq0.4)\approx0\).

Answer:

B. The result is highly unlikely because its probability, \(0.0000\), is less than \(0.05\). Thus, the sample is not a good representative sample.