Identify the claim and state H0 and Ha: (b) find the...

Identify the claim and state $H_{0}$ and $H_{a:}$ (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. If convenient, use technology. A research center claims that more than $80 \%$ of females ages $20-29$ are taller than 62 inches. In a random sample of 150 females ages $20-29,79 \%$ are taller than 62 inches. At $\alpha=0.10,$ is there enough evidence to support the center's claim?

Identify the claim and state $H_{0}$ and $H_{a:}$ (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. If convenient, use technology.
A research center claims that more than $80 \%$ of females ages $20-29$ are taller than 62 inches. In a random sample of 150 females ages $20-29,79 \%$ are taller than 62 inches. At $\alpha=0.10,$ is there enough evidence to support the center's claim?

Key Concepts

Decision and Interpretation

The decision in a hypothesis test is based on whether the calculated test statistic falls within the rejection region defined by the critical value(s). After making a decision, the conclusion is interpreted in the context of the original claim, indicating whether the evidence from the sample supports or contradicts the claim about the population parameter.

Test Statistic Calculation

The test statistic quantifies how far the observed sample statistic is from the hypothesized population parameter, measured in terms of standard error units. In the context of a proportion test, a z-test statistic is typically calculated, which then is compared to the critical value from the standard normal distribution to determine whether to reject the null hypothesis.

Critical Value and Rejection Region

The critical value is a threshold derived from the chosen significance level (alpha) that defines the boundary or boundaries for the rejection region in a hypothesis test. The rejection region is the set of values for the test statistic that would lead to the rejection of the null hypothesis if the observed test statistic falls within that region.

Null and Alternative Hypotheses

The null hypothesis (H0) represents the default or status quo statement regarding the population parameter, while the alternative hypothesis (Ha) represents the claim or assertion that contradicts the null hypothesis. In hypothesis testing, formulating these hypotheses clearly is crucial because they lay the foundation for the subsequent statistical analysis, determining whether the evidence supports the claim being made.

Hypothesis Testing

Hypothesis testing is a statistical method that involves making an assumption about a population parameter and then using sample data to assess the evidence against that assumption. It is used to decide whether there is enough evidence to reject a stated hypothesis or accept it as plausible, based on the probability of observing the sample data if the hypothesis were true.

Sampling Distribution of the Proportion

When testing a hypothesis about a population proportion, the sample proportion is treated as a random variable that follows an approximately normal distribution, given that certain conditions (such as a sufficiently large sample size) are satisfied. This distribution allows statisticians to compute probabilities and make decisions based on the observed sample data.

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