Question

According to a survey in a country, 21% of adults do not own a credit card. Suppose a simple random sample of 300 adults is obtained. Complete parts (a) through (d) below. (a) Describe the sampling distribution of p, the sample proportion of adults who do not own a credit card. Choose the phrase that best describes the shape of the sampling distribution of p below. O A. Approximately normal because n ≤0.05N and np(1-p) > 10 OB. Approximately normal because n≤0.05N and np(1-p) < 10 O C. Not normal because n ≤ 0.05N and np(1-p) < 10 O D. Not normal because n≤ 0.05N and np(1-p) ≥ 10 Determine the mean of the sampling distribution of p. (Round to two decimal places as needed.) HA = p Determine the standard deviation of the sampling distribution of p. (Round to three decimal places as needed.) p (b) What is the probability that in a random sample of 300 adults, more than 24% do not own a credit card? The probability is (Round to four decimal places as needed.) Interpret this probability. If 100 different random samples of 300 adults were obtained, one would expect to result in more than 24% not owning a credit card. (Round to the nearest integer as needed.) (c) What is the probability that in a random sample of 300 adults, between 19% and 24% do not own a credit card? The probability is. (Round to four decimal places as needed.) Interpret this probability. Interpret this probability. If 100 different random samples of 300 adults were obtained, one would expect to result in between 19% and 24% not owning a credit card. (Round to the nearest integer as needed.) (d) Would it be unusual for a random sample of 300 adults to result in 57 or fewer who do not own a credit card? Why? Select the correct choice below and fill in the answer box to complete your choice. (Round to four decimal places as needed.) A. The result is unusual because the probability that p is less than or equal to the sample proportion is B. The result is unusual because the probability that p is less than or equal to the sample proportion is C. The result is not unusual because the probability that p is less than or equal to the sample proportion is O D. The result is not unusual because the probability that p is less than or equal to the sample proportion is which is greater than 5%. which is less than 5%. " which is less than 5%. which is greater than 5%.

Answer 1

The sampling distribution of the sample proportion of adults who do not own a credit card is approximately normal because np and nq are greater than five. The mean of this distribution is the same as the population proportion, 0.21, and its standard deviation is calculated using the formula for a proportion in a binomial distribution.

Step-by-step explanation:

To determine the shape of the sampling distribution of the sample proportion p of adults who do not own a credit card, we apply the conditions for a binomial distribution approximated by a normal distribution, which requires that both np and nq are greater than five. Given the sample size n of 300 and the proportion p of 0.21, we see that np = 300×0.21 = 63 and n(1-p) = 300×0.79 = 237, both are greater than five. Therefore, the correct answer is A. Approximately normal because np(1-p) > 10.

The mean μ of the sampling distribution of p is equal to the population proportion, which in this case is 0.21. The standard deviation σ of the sampling distribution of p is calculated using the formula σ = √p(1-p)/n, which gives σ = √(0.21)×0.79/300 = 0.022.

Answer 2

(a) The sampling distribution of pis approximately normal n > 30 with a mean
`(\(\bar{p}\))` of 0.21 and standard deviation
`(\(\sigma_{\bar{p}}\))` of 0.025. (b) The probability of more than 24% adults not owning a credit card in a random sample of 300 is 0.0384, indicating an unusual event.

Step-by-step explanation:

(a) The shape of the sampling distribution is determined by the conditions n > 30, n ≤ 0.05N, and np(1-p) > 10. In this case, with n = 300, N being the population size, and p being the proportion of adults who do not own a credit card, all conditions are satisfied, leading to an approximately normal distribution.

(b) Probability calculations involve using the normal distribution with the mean and standard deviation of the sampling distribution. The result of 0.0384 suggests that observing more than 24% not owning a credit card is a relatively rare event, occurring around 4 times in 100 samples.

(c) Calculating the probability for a specific percentage range involves finding the area under the normal curve. The probability of 0.1619 indicates that obtaining a sample with a proportion between 19% and 24% not owning a credit card is more likely, occurring approximately 16 times in 100 samples.

(d) To determine if a result is unusual, the probability is compared to the significance level of 5%. With a probability of 0.0001, much less than 0.05%, it would be considered unusual to observe 57 or fewer adults not owning a credit card in a random sample of 300.