Question

a study was conducted to determine the proportion of people who dream in black and white instead of color. among 288 people over the age of 55, 64 dream in black and white, and among 299 people under the age of 25, 11 below. use a 0.05 significance level to test the claim that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25. complete parts (a) through (c)

a. test the claim using a hypothesis test
consider the first sample to be the sample of people over the age of 55 and the second sample to be the sample of people under the age of 25. what are the null and alternative hypotheses for the hypothesis test?
a. ( h_0: p_1
eq p_2 ) ( h_1: p_1 = p_2 )
b. ( h_0: p_1 = p_2 ) ( h_1: p_1
eq p_2 )
c. ( h_0: p_1 leq p_2 ) ( h_1: p_1
eq p_2 )
d. ( h_0: p_1 geq p_2 ) ( h_1: p_1
eq p_2 )
e. ( h_0: p_1 = p_2 ) ( h_1: p_1 > p_2 )
f. ( h_0: p_1 = p_2 ) ( h_1: p_1 < p_2 )
identify the test statistic.
z = 6.72
(round to two decimal places as needed.)
identify the p - value.
p - value =
(round to three decimal places as needed.)

Explanation:

Step1: Determine Hypotheses

The claim is that the proportion of people over 55 (\(p_1\)) who dream in black and white is greater than that for those under 25 (\(p_2\)). So, the null hypothesis (\(H_0\)) is \(p_1 = p_2\), and the alternative hypothesis (\(H_1\)) is \(p_1 > p_2\). This matches option E: \(H_0: p_1 = p_2\), \(H_1: p_1 > p_2\).

Step2: Find P - value for \(z = 6.72\)

For a right - tailed test (since \(H_1: p_1 > p_2\)) with \(z = 6.72\), we use the standard normal distribution. The P - value is the probability that \(Z>6.72\). Since the standard normal table typically goes up to \(z = 3.5\) or so, and for \(z = 6.72\), the area to the right of \(z = 6.72\) is extremely small. Using a calculator or software (like the normal - cdf function in a TI - 84, where normal - cdf(6.72, 1000, 0, 1)), we find that the P - value is approximately \(0.000\) (rounded to three decimal places).

Answer:

Part (a) Hypotheses:

E. \(H_0: p_1 = p_2\), \(H_1: p_1 > p_2\)

Part (b) P - value:

\(0.000\)