Question

A certain article indicates that in a sample of 1,000 dog owners, 680 said that they take more pictures of their dog than of their significant others or friends, and 490 said that they are more likely to complain to their dog than to a friend. Suppose that it is reasonable to consider this sample as representative of the population of dog owners.
(a) Construct a 90% confidence interval for the proportion of dog owners who take more pictures of their dog than of their significant others or friends. (Use a table or technology. Round your answers to three decimal places.)
(______),(_________)
(b) Construct a 95% confidence interval for the proportion of dog owners who are more likely to complain to their dog than to a friend. (Use a table or technology. Round your answers to three decimal places.)
(_______),(_______)

Answer 1

Answer: a) a 90% confidence interval for the proportion of dog owners who take more pictures of their dog than of their significant others or friends is (0.656,0.704).

b) a 95% confidence interval for the proportion of dog owners who are more likely to complain to their dog than to a friend is (0.459,0.521).

Step-by-step explanation:

Confidence interval for a population proportion is given by:-

$\hat{p}\pm z(\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}})$

, where $\hat{p}$ = sample proportion , n= sample size, z= critical z-value.

As per given,

a) n=1000

Sample proportion of dog owners say they take more pictures of their dog than of their significant others or friends =$\hat{p}=\dfrac{680}{1000}=0.68$

critical value for 90% confidence = 1.645  [By table]

A 90% confidence interval for the proportion of dog owners who take more pictures of their dog than of their significant others or friends.

$0.68\pm (1.645)\sqrt{\dfrac{0.68(1-0.68)}{1000}}\\\\\approx0.68\pm0.024=(0.68-0.024,0.68+0.024)=(0.656,0.704)$

Hence, a 90% confidence interval for the proportion of dog owners who take more pictures of their dog than of their significant others or friends is (0.656,0.704).

b) Sample proportion of dog owners say they are more likely to complain to their dog than to a friend =$\hat{p}=\dfrac{490}{1000}=0.49$

critical value for 95% confidence = 1.96  [By table]

A 95% confidence interval for the proportion of dog owners who are more likely to complain to their dog than to a friend:

$0.49\pm (1.96)\sqrt{\dfrac{0.49(1-0.49)}{1000}}\\\\\approx0.49\pm0.031=(0.49-0.031,0.49+0.031)=(0.459,0.521)$

Hence, a 95% confidence interval for the proportion of dog owners who are more likely to complain to their dog than to a friend is (0.459,0.521).