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University X has 20,000 students. Suppose Irah has drawn a simple random sample of 600 students...

Question:

University X has 20,000 students. Suppose Irah has drawn a simple random sample of 600 students from University X. In this sample, 240 students at University own Apple iPhones. At a 95% level of confidence, test the null hypothesis that not more than 7500 students at University X own Apple iPhones.

Confidence Interval

The confidence interval provides the range of value which mean that the value of true population paramter lies within the confidence interval at the given confidence level.

Answer and Explanation:

Given information

Sample size: 600

Number of students at University own Apple iPhones: 240


The value of sample proportion is given below.

{eq}\begin{align*} \hat P &= \dfrac{X}{n}\\ &= \dfrac{{240}}{{600}}\\ &= 0.4 \end{align*}{/eq}

The 95% confidence limit is calculated as follows.

{eq}\begin{align*} P\left( {\hat P - {Z_{\alpha /2}}\sqrt {\dfrac{{\hat P\left( {1 - \hat P} \right)}}{n}} < p < \hat P + {Z_{\alpha /2}}\sqrt {\dfrac{{\hat P\left( {1 - \hat P} \right)}}{n}} } \right) &= 0.95\\ P\left( {0.4 - 1.96\sqrt {\dfrac{{0.4\left( {1 - 0.4} \right)}}{{600}}} < p < 0.4 + 1.96\sqrt {\dfrac{{0.4\left( {1 - 0.4} \right)}}{{600}}} } \right) &= 0.95\\ P\left( {0.3608 < p < 0.4392} \right) &= 0.95 \end{align*}{/eq}

The value of true proportion is (7500/20000 = 0.375) which lies within the interval which mean the null hypothesis is failed to rejected and it is concluded that not more than 7500 students at University X own Apple iPhones.


Learn more about this topic:

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Finding Confidence Intervals with the Normal Distribution

from Statistics 101: Principles of Statistics

Chapter 9 / Lesson 3
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