Hypothesis testing for population proportion using one sample.

Here we are concerned with making an inference about a population proportion consisting of two categories(like Success or Failures, Pro-Villar or Anti-Villar) from data obtained in a sample of size $n$ .

The null hypothesis is that the population proportion $p_0 = \pi$ .

Mean and variance of sample proportion

From the binomial distribution, we know that the mean would be $\pi$ and the variance is $\pi (1 -\pi) /n$

In terms of counts X, the mean is $n\pi$ and the variance is $n\pi (1-\pi)$ .

An estimate obtained from a sample of size $n$ would be denoted by p (computed as X/n where X is the number of successes).

Standard error of estimated p

The standard error or standard deviation of the estimate of proportion is given by $\sqrt{\pi (1-\pi) /n}$

The normalized test statistic.

Assuming n is large enough, the normal approximation can be used and the test statistic would be given by

$Z_{test} = \frac{p- \pi}{\sqrt{\frac{\pi (1-\pi)} {n}}}$ .

In terms of counts, it is $Z_{test} = \frac{X - n\pi}{\sqrt{n\pi (1 - \pi)}}$

Confidence Intervals for population proportions.

Let $z_{crit}$ be the critical value(s) according to the level of significance $\alpha$ . For example the lower and critical critical values for a two sided test at the 5 percent level of significance is -1.96, and 1.96 but is 1.64 for a right single sided test.

Then for a two-sided test, we accept $H_0$ depending on whether $Z_{test} \in [-z_{crit}, z_{crit}]$ or not respectively.

For convenience, here are the acceptance regions or intervals for various common values of $\alpha$ .

$\alpha$	left sided test : $H_1:p_0 \le \pi$	double sided test: $H_1:p_0 \ne \pi$	right sided test: $H_1:p_0 \ge \pi$
0.01	[-2.36, $\infty$ )	[-2.58, 2.58]	( $-\infty$ , 2.36]
0.05	[-1.64, $\infty$ )	[-1.96, 1.96]	( $-\infty$ , 1.64]
0.10	[-1.28, $\infty$ )	[-1.64, 1.64]	( $-\infty$ , 1.28]

The p-value of the test.

Suppose that we obtained the sample proportion p. Convert this to a test statistic $Z_{test}$ . Then the p-value would be given as

Condition	Formula
$Z_{test} \ge 0$	$1-pnorm(Z_{test}$ )
$Z_{test} \le 0$	$pnorm(Z_{test}$ )

Here pnorm( $Z_{test}$ ) is the area from ( $-\infty, Z_test$ ) using the standard normal distribution. The double-sided test would have a p-value twice the p-values obtained for the single-sided tests.

The margin of error

The confidence interval corresponding to $1- \alpha)$ confidence level coefficient is centered on p, and has upper and lower confidence limits of $p \pm Z_{crit} (se).$ where se is the standard error of the sample proportion. The margin of error is $Z_{crit} (se).$ .

Examples here!! TBD.

Had a tough time sorting out the latex problems! It basically involves the ‘<‘ sign inside latex formulas. changed it to \le so the html display would be all right.

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