Math  /  Data & Statistics

QuestionAccording to a survey in a country, 39%39 \% of adults do not own a credit card. Suppose a simple random sample of 400 adults is obtained. Comple \qquad B. Not normal because n0.05 N\mathrm{n} \leq 0.05 \mathrm{~N} and np(1p)<10\mathrm{np}(1-\mathrm{p})<10 C. Not normal because n0.05 Nn \leq 0.05 \mathrm{~N} and np(1p)10\mathrm{np}(1-\mathrm{p}) \geq 10 D. Approximately normal because n0.05 Nn \leq 0.05 \mathrm{~N} and np(1p)10\mathrm{np}(1-\mathrm{p}) \geq 10
Determine the mean of n~\tilde{n} e sampling distribution of p^\hat{p}. μp^=.39\mu_{\hat{p}}=.39 (Round to two decimal places as needed.) Determine the standard deviation of the sampling distribution of p^\hat{p}. σp^=.024\sigma_{\hat{p}}=.024 (Round to three decimal places as needed.) (b) What is the probability that in a random sample of 400 adults, more than 42%42 \% do not own a credit card?
The probability is .1056 . (Round to four decimal places as needed.) Interpret this probability. If 100 different random samples of 400 adults were obtained, one would expect 11 to result in more than 42%42 \% not owning a credit card. (Round to the nearest integer as needed.) (c) What is the probability that in a random sample of 400 adults, between 34%34 \% and 42%42 \% do not own a credit card?
The probability is \square (Round to four decimal places as needed.)

Studdy Solution
Calculate the probability that between 34% 34\% and 42% 42\% do not own a credit card:
Convert 34% 34\% to a proportion:
p^1=0.34\hat{p}_1 = 0.34
Calculate the z-scores:
z1=p^1μp^σp^=0.340.390.0242.08z_1 = \frac{\hat{p}_1 - \mu_{\hat{p}}}{\sigma_{\hat{p}}} = \frac{0.34 - 0.39}{0.024} \approx -2.08
z2=p^2μp^σp^=0.420.390.0241.25z_2 = \frac{\hat{p}_2 - \mu_{\hat{p}}}{\sigma_{\hat{p}}} = \frac{0.42 - 0.39}{0.024} \approx 1.25
Find the probability P(0.34<p^<0.42) P(0.34 < \hat{p} < 0.42) :
P(2.08<Z<1.25)=P(Z<1.25)P(Z<2.08)P(-2.08 < Z < 1.25) = P(Z < 1.25) - P(Z < -2.08)
Using the standard normal distribution table or calculator:
P(Z<1.25)0.8944P(Z < 1.25) \approx 0.8944
P(Z<2.08)0.0188P(Z < -2.08) \approx 0.0188
P(2.08<Z<1.25)0.89440.0188=0.8756P(-2.08 < Z < 1.25) \approx 0.8944 - 0.0188 = 0.8756
The probability is:
0.8756\boxed{0.8756}

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