Determine whether it is appropriate to use the normal...

Determine whether it is appropriate to use the normal distribution to estimate the p-value. If it is appropriate, use the normal distribution and the given sample results to complete the test of the given hypotheses. Assume the results come from a random sample and use a $5 \%$ significance level. Test $H_{0}: p=0.2$ vs $H_{a}: p \neq 0.2$ using the sample results $\hat{p}=0.26$ with $n=1000$

Determine whether it is appropriate to use the normal distribution to estimate the p-value. If it is appropriate, use the normal distribution and the given sample results to complete the test of the given hypotheses. Assume the results come from a random sample and use a $5 \%$ significance level.
Test $H_{0}: p=0.2$ vs $H_{a}: p \neq 0.2$ using the sample results $\hat{p}=0.26$ with $n=1000$

Key Concepts

Normal Approximation to the Binomial

When the sample size is large and specific conditions are met (typically, both np and n(1-p) are sufficiently high), the binomial distribution can be approximated by a normal distribution. This approximation simplifies calculations in hypothesis testing by allowing the use of z-scores and standard normal tables, leveraging the Central Limit Theorem which ensures that the distribution of the sample proportion is approximately normal for large n.

Hypothesis Testing for Proportions

This is a statistical method used to assess whether an observed sample proportion significantly deviates from a hypothesized proportion under the null hypothesis. The process involves computing the standard error based on the hypothesized proportion and sample size, then determining the test statistic, which is compared against a standard normal distribution to evaluate the likelihood of the observed outcome.

P-value Calculation

The p-value is the probability of observing a test statistic as extreme or more extreme than the one calculated from the sample, assuming the null hypothesis is true. It quantifies the evidence against the null hypothesis; a lower p-value indicates stronger evidence prompting rejection of the null hypothesis. In a two-tailed test, the p-value accounts for deviations in both directions from the hypothesized proportion.

Significance Level and Decision Rule

The significance level (often set at 5%) is the threshold used to decide whether to reject the null hypothesis. If the calculated p-value is lower than the significance level, the result is considered statistically significant, leading to the rejection of the null hypothesis. This threshold helps control the Type I error rate, which is the probability of incorrectly rejecting a true null hypothesis.

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