Questions: A 2003 study of dreaming found that out of a random sample of 117 people, 87 reported dreaming in color. However, the rate of reported dreaming in color that was established in the 1940 was 0.29. Check to see whether the conditions for using a one-proportion z-test are met assuming the researcher wanted to test to see if the proportion dreaming in color had changed since the 1940. Are the conditions met? A. No, the sample size is not large enough to produce at least 10 successes and 10 failures. B. Yes, all the conditions are met. C. No, the observations are not independent. D. No, the population is not more than 10 times bigger than the sample size.

A 2003 study of dreaming found that out of a random sample of 117 people, 87 reported dreaming in color. However, the rate of reported dreaming in color that was established in the 1940 was 0.29. Check to see whether the conditions for using a one-proportion z-test are met assuming the researcher wanted to test to see if the proportion dreaming in color had changed since the 1940.

Are the conditions met?
A. No, the sample size is not large enough to produce at least 10 successes and 10 failures.
B. Yes, all the conditions are met.
C. No, the observations are not independent.
D. No, the population is not more than 10 times bigger than the sample size.

Transcript text: A 2003 study of dreaming found that out of a random sample of 117 people, 87 reported dreaming in color. However, the rate of reported dreaming in color that was established in the 1940 was 0.29 Check to see whether the conditions for using a one-proportion $z$-test are met assuming the researcher wanted to test to see if the proportion dreaming in color had changed since the 1940 . Are the conditions met? A. No, the sample size is not large enough to produce at least 10 successes and 10 failures. B. Yes, all the conditions are met. C. No, the observations are not independent. D. No, the population is not more than 10 times bigger than the sample size.

Solution

Solution Steps

Step 1: Calculate Expected Successes and Failures

To determine if the conditions for using a one-proportion $ z $-test are met, we first calculate the expected number of successes and failures based on the hypothesized population proportion $ p_0 = 0.29 $ and the sample size $ n = 117 $.

The expected number of successes is given by: \[ E(S) = n \cdot p_0 = 117 \cdot 0.29 \approx 33.93 \]

The expected number of failures is calculated as: \[ E(F) = n \cdot (1 - p_0) = 117 \cdot (1 - 0.29) \approx 83.07 \]

Step 2: Check Conditions for the One-Proportion $ z $-Test

Next, we check if both expected successes and failures are at least 10:

$ E(S) \approx 33.93 \geq 10 $
$ E(F) \approx 83.07 \geq 10 $

Since both conditions are satisfied, we conclude that the sample size is large enough to produce at least 10 successes and 10 failures.

Step 3: Independence of Observations

We also assume that the sample observations are independent, which is generally valid if the sample is random and the population is at least 10 times larger than the sample size. Given that the sample size is 117, the population should be at least $ 1170 $ for this assumption to hold.