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 $H_{0}: p=0.3$ vs $H_{a}: p<0.3$ using the sample results $\hat{p}=0.21$ with $n=200$

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 $H_{0}: p=0.3$ vs $H_{a}: p<0.3$ using the sample results $\hat{p}=0.21$ with $n=200$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population parameter based on sample data. It involves formulating a null hypothesis, which represents a default or status?quo assumption, and an alternative hypothesis, which is considered if there is sufficient evidence to suggest a deviation from the null hypothesis. The process involves using a test statistic to decide whether the sample data sufficiently contradicts the null hypothesis in favor of the alternative.

Null and Alternative Hypotheses

The null hypothesis (H?) represents the assumption that there is no effect or no difference from a stated value, while the alternative hypothesis (H?) suggests a deviation from that value. In tests of proportions, the null hypothesis often states that the population proportion is equal to a specific value, and the alternative hypothesis may propose that it is greater, less, or simply different from that value, depending on the test direction (one-tailed or two-tailed).

Normal Approximation for Proportions

The normal approximation for proportions is used when the sampling distribution of the sample proportion can be well approximated by a normal distribution. This is typically justified by the Central Limit Theorem for sufficiently large samples. Specifically, the approximation holds if np and n(1-p) are both greater than or equal to a threshold (commonly 5 or 10), ensuring that the distribution of the sample proportion is not overly skewed and that the normal model is a reasonable assumption.

Sampling Distribution of the Sample Proportion

The sampling distribution of the sample proportion describes the probability distribution of the proportion estimates obtained from all possible random samples of a fixed size from the population. This distribution is centered at the true population proportion, and its standard deviation is given by the square root of p(1-p)/n when the sampling is from a binomial process. When conditions are met, this distribution is approximately normal, which simplifies the process of hypothesis testing.

P-value Interpretation

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a test statistic as extreme as or more extreme than the one observed. In hypothesis testing, it provides a measure of the strength of the evidence against the null hypothesis. A low p-value indicates that the observed data is unlikely under the null hypothesis, which may lead to rejecting the null in favor of the alternative hypothesis.

Significance Level

The significance level, often denoted by ?, is the threshold for deciding whether to reject the null hypothesis. It represents the probability of making a Type I error, which is the error of rejecting the null hypothesis when it is actually true. In practice, a significance level of 0.05 is commonly used, meaning there is a 5% risk of committing a Type I error when the null hypothesis is rejected.

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