Question

a study was conducted to determine the proportion of people who dream in black and white instead of color. among 306 people over the age of 55, 68 dream in black and white, and among 298 people under 25, 13 dream in black and white ( Based on data from do we dream in color) we want to use a 0.01 significance level to test the claim that the proportion of people over 55 who dream in black and white is greater than the proportion of those under 25.

Answer 1

Answer:

There is sufficient evidence to support the claim that the proportion of people over 55 who dream in black and white is greater than the proportion of those under 25. (0.1104, 0.2470)

Explanation:

Let :

$n_{1}$ be the number of people over the age of 55.

$n_{2}$ be the number of people under the age of 25.

$x_{1}$ be the number of people who dream in black and white over the age of 55.

$x_{2}$ is the number of people who dream in black and white under age of 25.

α be the significance level, from which the confidence level is calculated

Given :

$n_{1} = 306$

$n_{2} = 298$

$x_{1} = 68$

$x_{2} = 13$

$\alpha=0.01$

The sample proportion is the number of successes divided by the sample size:  

$\hat p_{1}=\frac{x_{1} }{n_{1} }=\frac{68}{306}\approx 0.2222$

$\hat p_{2}=\frac{x_{2} }{n_{2}}=\frac{13}{298} \approx 0.0436$

For confidence level 1 -  α = 0.99.

determine $\frac{z_{\alpha} }{2}=z_{0.005}$  using the normal probability table.

$\frac{z_{\alpha} }{2}=z_{0.005}=2.575$  

The margine error is then:  

$E=\sqrt[\frac{z_{\alpha} }{2}]{\frac{\hat p\hat q}{n_{1} }+\frac{\hat p\hat q}{n_{2}}}=\sqrt[2.575]{\frac{0.222(1-0.2222)}{306}+\frac{0.0436(1 - 0.044)}{298}}\approx. 0.0684$  

The confidence interval is then :

$0.1104 = (0.2222 - 0.0436)-0.0684 = (\hat p_{1}-\hat p_{2})-E < p_{1}- p_{2} < ( \hat p_{1}-\hat p_{2})+E = (0.222-0.0436) + 0.0684 = 0.2470$  

Confidence interval does not contain 0 thus the null hypothesis is rejected and the claim supported. There is sufficient evidence to support the claim that the proportion of people over 55 who dream in black and white is greater than the proportion of those under 25.