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Softa [21]
3 years ago
10

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.
SAT
1 answer:
Allushta [10]3 years ago
6 0

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.

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<h3> The critical value approach? </h3>
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<h3>What is the standard error of the distribution?</h3>

The standard error of the distribution is calculated as follows:

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