Consider the following hypothesis test: H0=p ≥.75 Ha=p<.75 A sample of 300 items was

Consider the following hypothesis test: H0=p ≥.75 ...

Consider the following hypothesis test: $$ \begin{aligned} & H_0=p \geq .75 \\ & H_{\mathrm{a}}=p<.75 \end{aligned} $$ A sample of 300 items was selected. Compute the $p$-value and state your conclusion for each of the following sample results. Use $\alpha=.05$. a. $\bar{p}=.68$ c. $\vec{p}=.70$ b. $\bar{p}=.72$ d. $\tilde{p}=.77$

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0=p \geq .75 \\
& H_{\mathrm{a}}=p<.75
\end{aligned}
$$
A sample of 300 items was selected. Compute the $p$-value and state your conclusion for each of the following sample results. Use $\alpha=.05$.
a. $\bar{p}=.68$
c. $\vec{p}=.70$
b. $\bar{p}=.72$
d. $\tilde{p}=.77$

Key Concepts

One-tailed Test

A one-tailed test is used when the alternative hypothesis specifies a direction of the effect. In a one-tailed test, the rejection region for the test statistic is all in one tail of the distribution, focusing on deviations in a specific direction rather than in both directions.

p-value

The p-value is the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is correct. It quantifies the strength of the evidence against the null hypothesis; a smaller p-value indicates stronger evidence to reject the null hypothesis.

Test Statistic for Proportions

For tests involving proportions, the test statistic is typically calculated using the z-test when sample sizes are large. It measures the number of standard errors that the observed sample proportion is away from the hypothesized population proportion, allowing for the computation of the p-value using the standard normal distribution.

Null and Alternative Hypotheses

The null hypothesis (H?) is a statement about the population parameter that indicates no effect or no difference, while the alternative hypothesis (H?) represents the research claim or the effect in question. In hypothesis testing, the null hypothesis is assumed to be true until the evidence provided by the data suggests otherwise.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds true for the entire population. It involves setting up a null hypothesis, which represents a default position, and an alternative hypothesis, which reflects what the researcher is trying to support.

Significance Level

The significance level, denoted by ?, is the threshold for rejecting the null hypothesis. It represents the probability of making a Type I error (rejecting a true null hypothesis). Common significance levels include 0.05, 0.01, and 0.10, which help determine whether the observed data are sufficiently unlikely under the assumption that the null hypothesis is true.

Transcript

0:00 Hello.

00:01 So here the hypotheses to be tested are the null hypothesis is that p is greater than equal to 0 .75.

00:07 And the alternative hypothesis is that p is less than 0 .75.

00:11 So here we have that our sample size n is 300.

00:14 And our sample proportion, p bar is equal to 0 .68.

00:20 So the test statistic then is given by p bar minus p not.

00:25 So that's going to be 0 .68 .8.

00:32 Minus 0 .75 and then divided by the square root of p .knot times one minus p .knot all over n.

00:41 So that's going to be 0 .75 times 1 minus 0 .75, all divided by 300.

00:51 And this should give us negative 2 .8.

00:55 So then the probability that z is less than or equal to z equals negative 2 .8.

01:00 That is the area under the standard normal curve to the left of z equals negative 2 .8 is obtained from our table and we get the p value is going to be equal to 0 .0026 given that alpha is equal to 0 .05 we have that our p value is going to be less than alpha so therefore we can reject the null hypothesis at a 5 % level of significance and then conclude that p is less than 0 .75.

01:38 So we accept the alternative hypothesis.

01:41 And then for part b, the hypotheses, again, p greater than equal to 0 .75, alternative p less than 0 .75.

01:50 So sample proportion then is now, again, 300...

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