Question

A total of 100 rats whose mothers were exposed to high levels of tobacco smoke during pregnancy were put through a simple maze. The maze required the rats to make a choice between going left or right at the outset. Eighty of the rats went right when running the maze for the first time. Assume that the 100 rats can be considered an SRS from the population of all rats born to mothers exposed to high levels of tobacco smoke during pregnancy. (Note that this assumption may or may not be reasonable, but researchers often assume lab rats are representative of such larger populations because they are often bred to have very uniform characteristics.) Let p be the proportion of rats in this population that would go right when running the maze for the first time. A 90% confidence interval for p is

A. 0.8 ± 0.040.
B. 0.8 ± 0.066.
C. 0.8 ± 0.078.

Answer 1

The 90% confidence interval for the proportion \( p \) of rats born to mothers exposed to high levels of tobacco smoke, based on a sample of 100 rats, is approximately 0.734 to 0.866, making option B the closest match.

To calculate a 90% confidence interval for the population proportion p of rats born to mothers exposed to high levels of tobacco smoke during pregnancy, we can use the formula for the confidence interval:

`\[ \text{Confidence Interval}`= \
`hat{p} \pm z * \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]`

Where:

-p is the sample proportion (proportion of rats going right), z is the critical value for a 90% confidence interval (which is approximately 1.645 for a large sample size),

n is the sample size.

Given that p =
`(80)/(100) = 0.8 \),`
`\( z \approx 1.645 \)`, and n = 100 , we can plug these values into the formula:

Confidence Interval=
`0.8 \pm 1.645 * \sqrt{(0.8 * (1 - 0.8))/(100)} \]`

Now, calculate the standard error:

SE =
`\sqrt{(0.8 * (1 - 0.8))/(100)} \]`

`\[ SE = \sqrt{(0.8 * 0.2)/(100)} \]`

`\[ SE = \sqrt{(0.16)/(100)} \]`

`\[ SE = √(0.0016) \]`

SE = 0.04

Now, plug the values into the confidence interval formula:

Confidence Interval
`= 0.8 \pm 1.645 * 0.04 \]`

Confidence Interval =
`0.8 \pm 0.0658 \]`

Therefore, the 90% confidence interval for
`\( p \) is approximately \( 0.7342 \)`to
`\( 0.8658 \).`

Comparing this with the given options:

B.
`\( 0.8 \pm 0.066 \)` is the closest match.