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 $H_{0}: p=0.25$ vs $H_{a}: p<0.25$ using the sample results $\hat{p}=0.16$ with $n=100$

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 $H_{0}: p=0.25$ vs $H_{a}: p<0.25$ using the sample results $\hat{p}=0.16$ with $n=100$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves stating a null hypothesis and an alternative hypothesis, computing a test statistic from the data, and then determining whether the observed statistic is consistent with the null hypothesis.

Test of Proportion

A test of proportion specifically evaluates whether the proportion observed in a sample differs significantly from a pre-specified proportion of the population. This type of test is common in categorical data analysis and is used to determine if there is evidence to support a deviation from a hypothesized proportion.

Normal Approximation to the Binomial Distribution

When the sample size is large, the sampling distribution of a proportion can be approximated by a normal distribution, thanks to the central limit theorem. This approximation allows us to calculate a z statistic using the mean and standard error based on the null hypothesis proportion, making it easier to compute p-values for hypothesis testing.

Significance Level

The significance level, denoted by alpha (?), is the threshold for rejecting the null hypothesis. It represents the probability of making a Type I error—that is, rejecting a true null hypothesis. A 5% significance level means that there is a 5% risk of committing this error.

p-value

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one computed from the sample data under the assumption that the null hypothesis is true. If the p-value is less than the predetermined significance level, it indicates that the observed result is unlikely under the null hypothesis, leading to its rejection.

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