9.3: The Central Limit Theorem for Sample Proportions

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Page ID: 105852

Anton Butenko
Mt. San Jacinto College

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Theorem $\PageIndex{1}$ Central Limit Theorem for Sample Proportions

If $p$ is the proportion of a population that possesses a certain feature, then $\hat{p}$, the proportion of a sufficiently large sample of size $n$ that possess the same feature, is approximately normally distributed with the mean $\mu_{\hat{p}}=p$ and the standard deviation $\sigma_{\hat{p}}=\sqrt{\dfrac{p(1-p)}{n}}$. In this context, a "large" sample is such that $np\geq10$ and $n(1-p)\geq10$.

Symbolically, the statement can be written in the following way:

For $X\sim?(p)$, when $np\geq10$ and $n(1-p)\geq10$,

$\hat{p}\sim N\left(\mu_{\hat{p}}=p, \sigma_{\hat{p}}=\sqrt{\dfrac{p(1-p)}{n}}\right)$

Try It Yourself! $\PageIndex{0.1}$

Try It Yourself! $\PageIndex{0.2}$

Example $\PageIndex{1}$

According to a study, 5.4% of the passengers fly first-class. A random sample of size 215 was obtained by a researcher. Find the probability that at least 10 people in the sample are first-class passengers.

$Image of passengers waiting at the airport.$ Solution

Let $p$ be the proportion of people that fly first-class, then $p=0.054$.

Let $\hat{p}$ be the proportion of people that fly first-class in the sample of 215 random passengers. Since the sample is large, i.e.

$np=215\cdot0.054=11.61\geq10$ and $n(1-p)=215\cdot(1-0.054)=215\cdot0.946=203.39\geq10$

by CLT,

$\hat{p}\sim N\left(\mu_{\hat{p}}=p=0.054, \sigma_{\hat{p}}=\sqrt{\dfrac{p(1-p)}{n}}=\sqrt{\dfrac{0.054(1-0.054)}{215}}=0.0154\right)$

Then the probability that at least 10 people (i.e., proportion is at least 10/215) in the sample are first-class passengers can be expressed as

$P(\hat{p}>\frac{10}{215})=P(\hat{p}>0.0465)=P\left(\dfrac{\hat{p}-\mu_{\hat{p}}}{\sigma_{\hat{p}}}>\dfrac{0.0465-0.054}{0.0154}\right)=P(Z>-0.49)=0.6879=68.79\%$

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Theorem \(\PageIndex{1}\) Central Limit Theorem for Sample Proportions

Try It Yourself! \(\PageIndex{0.1}\)

Try It Yourself! \(\PageIndex{0.2}\)

Example \(\PageIndex{1}\)

Try It Yourself! \(\PageIndex{1}\)

Try It Yourself! \(\PageIndex{2}\)

Try It Yourself! \(\PageIndex{3}\)

Try It Yourself! \(\PageIndex{4}\)

Try It Yourself! \(\PageIndex{5}\)