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 $H_{0}: p=0.8$ vs $H_{a}: p>0.8$ using the sample results $\hat{p}=0.88$ with $n=50$

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 $H_{0}: p=0.8$ vs $H_{a}: p>0.8$ using the sample results $\hat{p}=0.88$ with $n=50$

Key Concepts

Significance Level

The significance level, denoted by alpha (?), is the threshold for rejecting the null hypothesis in hypothesis testing. Common levels, such as 0.05, represent a 5% risk of wrongly rejecting the null hypothesis. It provides an objective criterion to decide whether the observed result is statistically significant.

One-Tailed Hypothesis Test

This concept refers to hypothesis tests where the alternative hypothesis specifies a direction of the deviation from the null hypothesis (for example, p > 0.8). In these tests, the p-value is calculated for one tail of the distribution, and the decision to reject the null hypothesis is based on whether the p-value is less than the chosen significance level.

p-value

The p-value represents the probability of obtaining a test statistic as extreme as, or more extreme than, the one computed from the sample data, assuming that the null hypothesis is true. In the context of a one-tailed test, it quantifies the likelihood of observing a sample proportion at least as high (or low) as the one obtained in the direction specified by the alternative hypothesis.

Normal Approximation to the Binomial

This concept involves using the normal distribution as an approximation for a binomial distribution when certain conditions are met. Specifically, it is appropriate when both np and n(1-p) are large enough (commonly at least 10), ensuring that the distribution of the sample proportion is approximately normal, which is a key assumption for applying z-tests in hypothesis testing.

Central Limit Theorem

The Central Limit Theorem states that the distribution of the sample means (or sample proportions) approaches a normal distribution as the sample size increases, regardless of the population's distribution. This theorem justifies the normal approximation in hypothesis testing when dealing with large samples, allowing the use of z-scores to evaluate the test statistic.

Test Statistic

The test statistic in hypothesis testing involving a proportion is calculated using the formula z = (observed proportion - hypothesized proportion) divided by the standard error of the proportion. The standard error is computed as the square root of [p*(1-p)/n], where p is the proportion under the null hypothesis. This z-score helps in determining how many standard deviations the observed result is from the hypothesized value.

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