Question

it is reported that 28% of all adults own at least one houseplant. imagine that we survey a random sample of ( n = 400 ) adults and ask each adult to tell us if they own at least one houseplant. we know, if the sampling method were to be repeated many times, the proportion of adults who answer \yes\ to this question will naturally vary from sample to sample. in fact, if we look at the resulting sampling distribution, it will have a shape that is normal, a mean of 0.28, and a standard deviation of 0.022. between what two values will approximately 68% of the sample proportions in this distribution fall?

0.133 and 0.427
0.278 and 0.282
0.246 and 0.313
0.270 and 0.290
0.258 and 0.302

Explanation:

Step1: Recall the empirical rule for normal distributions

The empirical rule states that for a normal distribution, approximately 68% of the data lies within 1 standard deviation of the mean.

Step2: Calculate the lower and upper bounds

The mean ($\mu$) is 0.28 and the standard deviation ($\sigma$) is 0.022.
The lower bound is $\mu-\sigma=0.28 - 0.022=0.258$.
The upper bound is $\mu+\sigma=0.28+ 0.022 = 0.302$.

Answer:

0.258 and 0.302