Margin of Error & Hypothesis Tests

Section 1 of 3

Margin of Error

From the sample size

For a sample of size $n$, the margin of error is $E = \dfrac{1}{\sqrt{n}}$ — a bigger sample gives a smaller margin.

Worked example — several samples

Find the margin of error for samples of size $50$, $200$, $1001$, $2000$ and $97$.

$50$: $\tfrac{1}{\sqrt{50}} = 0.14 = 14\%$; $200$: $0.07 = 7\%$

$1001$: $0.03 = 3\%$; $2000$: $0.02 = 2\%$; $97$: $0.1 = 10\%$

$14\%, 7\%, 3\%, 2\%, 10\%$

Section 2 of 3

Confidence Interval

The 95% interval

Sample proportion $= \hat{p}$; population proportion $= p$. The confidence interval is $\hat{p} - \dfrac{1}{\sqrt{n}} \le p \le \hat{p} + \dfrac{1}{\sqrt{n}}$.

Worked example — form the interval

A sample of $n = 25$ found $0.36$ like a product. Find the margin of error and the confidence interval.

$E = \tfrac{1}{\sqrt{25}} = 0.2$

$0.36 - 0.2$ to $0.36 + 0.2$

$0.18 \le p \le 0.56$

You try

$30$ out of $120$ failed a test. Find a confidence interval for the population.

$\hat{p} = \tfrac{30}{120}$; $E = \tfrac{1}{\sqrt{120}}$; then $\hat{p} \pm E$.

$\hat{p} = 0.25$, $E = \tfrac{1}{\sqrt{120}} = 0.09$

$0.25 - 0.09$ to $0.25 + 0.09$

$0.16 \le p \le 0.34$

Section 3 of 3

Hypothesis Testing

The five steps

1. State $H_0$ (the claim) and $H_1$ (the opposite). 2. Find $\hat{p}$ (sample proportion). 3. Find $E = \tfrac{1}{\sqrt{n}}$. 4. Build the interval $\hat{p} - E \le p \le \hat{p} + E$. 5. If the claim is inside the interval, fail to reject $H_0$; if outside, reject $H_0$.

Worked example — Mary’s frees

Mary claims she scores $75\%$ of her frees. She takes $112$ and scores $72$. Test her claim.

$H_0$: scores $75\%$; $H_1$: not $75\%$

$\hat{p} = \tfrac{72}{112} = 64\%$; $E = \tfrac{1}{\sqrt{112}} = 9\%$

Interval: $55\% \le p \le 73\%$

$75\%$ is outside → reject $H_0$: she does not score $75\%$

reject $H_0$ — not $75\%$

Worked example — beauty company

A company claims $80\%$ of customers look younger. $76$ of $95$ agree. Test the claim.

$\hat{p} = \tfrac{76}{95} = 80\%$; $E = \tfrac{1}{\sqrt{95}} = 10\%$

Interval: $70\% \le p \le 90\%$

$80\%$ is inside → fail to reject $H_0$: $80\%$ are happy

fail to reject — $80\%$ holds

You try

A dog-food company claims $48\%$ of dogs prefer their food. $75$ of $136$ dogs prefer it. Test the claim.

$\hat{p} = \tfrac{75}{136}$; $E = \tfrac{1}{\sqrt{136}}$; is $48\%$ inside the interval?

$\hat{p} = 55\%$; $E = 0.085 = 9\%$

Interval: $46\% \le p \le 64\%$

$48\%$ is inside → fail to reject $H_0$: $48\%$ prefer it

fail to reject — $48\%$ holds

You try

A drug company claims a vaccine works for $72\%$. Of $273$ tested, $168$ were immune. Test the claim.

$\hat{p} = \tfrac{168}{273}$; $E = \tfrac{1}{\sqrt{273}}$; is $72\%$ inside?

$\hat{p} = 0.615 = 62\%$; $E = 6\%$

Interval: $56\% \le p \le 68\%$

$72\%$ is outside → reject $H_0$: it does not work for $72\%$

reject $H_0$ — not $72\%$

That’s Statistics — the whole topic.

Margin of error $\tfrac{1}{\sqrt{n}}$, confidence intervals $\hat{p} \pm \tfrac{1}{\sqrt{n}}$, and the five-step hypothesis test — every example from your notes.