Question

what is the probability of obtaining x = 116x=116 or fewer individuals with the characteristic? that is, what is the value of p(\hat{p}\le 0.58)p( p ^ ≤0.58)? (round to four decimal places)

Answer 1

Explanation:

To calculate the probability of obtaining x = 116 or fewer individuals with the characteristic, we need to use the normal distribution. We can use the sample proportion p-hat and the population proportion p to find the mean and standard deviation of the sampling distribution of p-hat.

We are given that n = 200 and p-hat = 0.58. The population proportion p is not given, so we will assume it to be 0.5 (which is a common assumption when the population proportion is unknown).

The mean of the sampling distribution of p-hat is:

μp-hat = p = 0.5

The standard deviation of the sampling distribution of p-hat is:

σp-hat = sqrt[p(1-p)/n] = sqrt[(0.5)(0.5)/200] = 0.0354

Now we can standardize the value of x using the formula:

z = (x - μp-hat) / σp-hat

For x = 116, we have:

z = (116 - 100*0.58) / 0.0354 = 1.69

Using a standard normal distribution table or calculator, we can find the probability of obtaining z ≤ 1.69 to be 0.9535. Therefore, the probability of obtaining x = 116 or fewer individuals with the characteristic is approximately 0.9535 (rounded to four decimal places).