Consider the following hypothesis test: H0: p=.20 Ha=p+.20 A sample of 400 provided

step 1 For the first part of this question, we're going to be determining the value of our Z, which can

Consider the following hypothesis test: H0: p=.20 ...

Consider the following hypothesis test: $$ \begin{aligned} & H_0: p=.20 \\ & H_{\mathrm{a}}=p+.20 \end{aligned} $$ A sample of 400 provided a sample proportion $\tilde{p}=.175$. a. Compute the value of the test statistic. b. What is the $p$-value? c. At $a=.05$, what is your conclusion? d. What is the rejection rule using the critical value? What is your conclusion?

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0: p=.20 \\
& H_{\mathrm{a}}=p+.20
\end{aligned}
$$
A sample of 400 provided a sample proportion $\tilde{p}=.175$.
a. Compute the value of the test statistic.
b. What is the $p$-value?
c. At $a=.05$, what is your conclusion?
d. What is the rejection rule using the critical value? What is your conclusion?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to assess evidence provided by sample data in favor of or against a specific claim or hypothesis about a population parameter. It involves formulating a null hypothesis (H0) that represents the status quo or a baseline assumption and an alternative hypothesis (Ha) which reflects the claim to be tested. Decisions are made based on the probability of observing a sample statistic under the null hypothesis, ensuring a structured approach to drawing inferences about the population.

Test Statistic

The test statistic is a standardized value computed from sample data that measures the degree of divergence from what is expected under the null hypothesis. In the context of proportions, it is typically calculated using the formula z = (p? - p0) / ?[p0(1-p0)/n], where p? is the observed sample proportion, p0 is the hypothesized proportion under H0, and n is the sample size. This statistic allows researchers to compare the observed outcome with the expected distribution assuming the null hypothesis is true.

p-value

The p-value represents the probability, assuming the null hypothesis is true, of obtaining a test statistic as extreme as or more extreme than the one computed from the sample data. It quantifies the strength of evidence against the null hypothesis. A small p-value indicates that such an extreme outcome is unlikely under H0 and provides grounds for rejecting the null hypothesis, whereas a larger p-value suggests that the observed data are consistent with the null hypothesis.

Significance Level

The significance level, denoted by ?, is a pre-determined threshold used to evaluate the p-value and decide whether to reject the null hypothesis. It represents the maximum probability of making a Type I error, which is the error of incorrectly rejecting a true null hypothesis. Common choices for ? include 0.05 or 0.01, and if the p-value is less than or equal to ?, the null hypothesis is rejected in favor of the alternative.

Rejection Rule and Critical Value

The rejection rule is a decision criterion that defines the set of values for the test statistic for which the null hypothesis will be rejected. This rule is often expressed in terms of a critical value, which is determined by the significance level and the sampling distribution of the test statistic. If the computed test statistic falls into the rejection region (e.g., less than the critical value in a one-tailed test), the null hypothesis is rejected; otherwise, it is not rejected. This approach provides a clear cut-off point for decision-making in hypothesis testing.

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