Question

A university officer wants to know the proportion of registered students that spend more than 20 minutes to get to school. He selects two parking decks at the university and talked with students at those decks. Of 25 students found in the decks, 12 have a commute under 20 minutes and 13 have a commute more than 20 minutes.
a. Yes.
b. No, because it wasn't a random sample.
c. No, because n(p-hat) < 10 or n(q-hat) < 10.
d. No, because the sample size wasn’t at least 30 and the population wasn’t normally distributed.
e. No, because we already know the population proportion.

Answer 1

Answer:

1) We need a random sample. For this case we assume that the sample selected was obtained using the simple random sampling method.

2) We need to satisfy the following inequalities:

$n\hat p =25*0.52= 13 \geq 10$

$n(1-\hat p) = 25*(1-0.52) =12 \geq 10$

So then we satisfy this condition

3) 10% condition. For this case we assume that the random sample selected n represent less than 10% of the population size N . And for this case we can assume this condition.

So then since all the conditions are satisfied we can conclude that we can apply the normal approximation given by:

$p \sim N (\hat p, \sqrt{\frac{\hat p (1-\hat p)}{n}})$

So then the answer for this case would be:

a. Yes.

Explanation:

For this case we assume that the question is: If in the experiment described we can use the normal approximation for the proportion of interest.

For this case we have a sample of n =25

And we are interested in the proportion of registered students that spend more than 20 minutes to get to school.

X = 13 represent the number of students in the sample selected that have a time more than 20 min.

And then the estimated proportion of interest would be:

$\hat p = \frac{X}{n}= \frac{13}{25}= 0.52$

And we want to check if we can use the normal approximation given by:

$p \sim N (\hat p, \sqrt{\frac{\hat p (1-\hat p)}{n}})$

So in order to do this approximation we need to satisfy some conditions listed below:

1) We need a random sample. For this case we assume that the sample selected was obtained using the simple random sampling method.

2) We need to satisfy the following inequalities:

$n\hat p =25*0.52= 13 \geq 10$

$n(1-\hat p) = 25*(1-0.52) =12 \geq 10$

So then we satisfy this condition:

3) 10% condition. For this case we assume that the random sample selected n represent less than 10% of the population size N . And for this case we can assume this condition.

So then since all the conditions are satisfied we can conclude that we can apply the normal approximation given by:

$p \sim N (\hat p, \sqrt{\frac{\hat p (1-\hat p)}{n}})$

So then the answer for this case would be:

a. Yes.