Question

question 2
1 pts
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
question 3
1 pts
return to the information presented in question 2. what is the probability that if we survey a random sample of 400 adults, the proportion who own at least one houseplant will be 0.245 or smaller?
0.0350
0.0548
0.1841
0.0227
at least 0.5000

Explanation:

Step1: Recall the empirical rule

For a normal distribution, approximately 68% of the data lies within \( \mu\pm\sigma \), where \( \mu \) is the mean and \( \sigma \) is the standard deviation.

Step2: Calculate the lower and upper bounds

Given \( \mu = 0.28 \) and \( \sigma=0.022 \).
The lower bound is \( \mu - \sigma=0.28 - 0.022=0.258 \).
The upper bound is \( \mu+\sigma = 0.28+0.022 = 0.302 \).

for Question 3:

Step1: Calculate the z - score

The formula for the z - score is \( z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}} \). Here, \( p = 0.28 \), \( \hat{p}=0.245 \), and \( n = 400 \). First, calculate the standard deviation \( \sigma=\sqrt{\frac{p(1 - p)}{n}}=\sqrt{\frac{0.28\times(1 - 0.28)}{400}}=\sqrt{\frac{0.28\times0.72}{400}}=\sqrt{\frac{0.2016}{400}}=\sqrt{0.000504}\approx0.022 \). Then, \( z=\frac{0.245 - 0.28}{0.022}=\frac{- 0.035}{0.022}\approx - 1.59 \).

Step2: Find the probability using the standard normal table

Looking up \( z=-1.59 \) in the standard normal table (or using a calculator with a normal - distribution function \( P(Z\leq z) \)), we find that \( P(\hat{P}\leq0.245)=P(Z\leq - 1.59)\approx0.0548 \).

Answer:

0.258 and 0.302