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 $H_{0}: p=0.75$ vs $H_{a}: p \neq 0.75$ using the sample results $\hat{p}=0.69$ with $n=120$

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 $H_{0}: p=0.75$ vs $H_{a}: p \neq 0.75$ using the sample results $\hat{p}=0.69$ with $n=120$

Key Concepts

Standard Error of a Proportion

The standard error measures the variability of the sample proportion by quantifying the expected difference between the sample proportion and the true population proportion. It is calculated using the formula sqrt[p*(1-p)/n], where p is the hypothesized proportion and n is the sample size, serving as a key component in constructing the test statistic for hypothesis testing.

p-value

The p-value is the probability, under the assumption of the null hypothesis, of obtaining a test statistic at least as extreme as the one observed. It is used to assess the strength of the evidence against the null hypothesis: a small p-value suggests that the observed data are unlikely under the null hypothesis.

Two-Tailed Test

A two-tailed test is used when the alternative hypothesis does not specify a direction of the effect; it simply indicates that a difference exists. This requires evaluating both tails of the distribution and comparing the test statistic against critical values on both sides, making it appropriate for testing hypotheses like p not equal to a specified value.

Hypothesis Testing

This is a statistical method used to evaluate evidence from a sample about a population parameter. It involves formulating a null hypothesis (a claim of no effect or status quo) and an alternative hypothesis (a claim of an effect or difference), and determining whether sample evidence is strong enough to reject the null in favor of the alternative based on a predetermined significance level.

Normal Approximation

This concept refers to the use of the normal distribution as an approximation for a binomial distribution when sample sizes are large enough. Due to the Central Limit Theorem, when the number of trials is sufficiently large, the distribution of the sample proportion can be approximated by a normal distribution, which simplifies computation of probabilities and critical values.

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