Perform the test of hypotheses indicated, using the data given. Use the critical value approach.

Perform the test of hypotheses indicated, using the data...

Key Concepts

p-value

The p-value is the probability of obtaining a test statistic at least as extreme as the one computed from the sample data, assuming that the null hypothesis is true. It provides a measure of the strength of the evidence against the null hypothesis. A small p-value indicates strong evidence in favor of rejecting the null hypothesis, and it is often compared with a predetermined significance level to make a decision in hypothesis testing.

Large Sample Tests

Large sample tests utilize the central limit theorem, which states that the sampling distribution of the sample mean (or other test statistics) approximates a normal distribution when the sample size is sufficiently large. This allows for the application of Z-tests or other asymptotic methods, simplifying the process of hypothesis testing as the critical values and p-values can be accurately determined using normal distribution approximations.

Hypothesis Testing

Hypothesis testing is a systematic method for evaluating a claim or hypothesis about a population parameter based on sample data. It involves formulating a null hypothesis (which represents a default or no-effect statement) and an alternative hypothesis (indicating the presence of an effect or difference). The process includes selecting an appropriate test statistic, determining its distribution under the null hypothesis, and then using this information to infer whether the observed data are consistent with the null hypothesis or whether there is sufficient evidence to support the alternative hypothesis.

Critical Value Approach

The critical value approach is a method in hypothesis testing where a threshold value (or values) is determined from the sampling distribution corresponding to a chosen significance level. When the test statistic computed from the sample data falls into the critical region defined by these critical values, the null hypothesis is rejected. This approach provides a clear rule for decision-making based on the probability of observing extreme values under the null hypothesis.

Transcript

00:01 The following is the solution to number six, and we have a couple hypothesis tests here where we're testing the difference is 0 .05 or the difference of the two is greater than 0 .05, given this following data, so 1100 are sample sizes, and then our p1 had is 0 .57, and our p2 had is 0 .48.

00:17 We're testing this at the 5 % level of significance.

00:20 So we're going to find that test statistic first, so z is equal to the difference, the observed differences, which would be .57 minus .48 minus the hypothesized difference, which is .05, and then we're dividing that by big square root, the p1 hat times one minus p1 hat, which would be .43 in this case, divided by that sample size, and then plus p2 hat .48 times one minus p2 hat, which would be .52, and divided by that sample size.

01:02 Whenever you plug that in, you should get negative 2 .606 as your test statistic.

01:09 Now to find the critical value, we look at two things.

01:13 We look at the alpha and we look at the tail of the test.

01:15 So this is a right tail test at alpha equals .05.

01:18 So if i go to second distribution and go to inverse norm, i can type in .05.

01:25 And then since this is a right tail test, i'll go to the right, and then i paste that rascal in there and i get 1 .645.

01:31 So 1 .645 is my z star.

01:36 So using the critical region method, i'm sorry, this is wrong.

01:48 I was looking at the wrong problem.

01:50 It's positive, and the reason why i had to stop there for a second is because it's greater than.

01:55 Sorry, it's 1 .886, so double -check my work there, 1 .886.

01:59 So anything to the right of 1 .645, i would reject, and this test statistic lands to the right.

02:07 Right, which means i reject h0.

02:10 So reject h0.

02:14 Now i also need to find the p value.

02:17 And the p value, and there we go, is the probability that i get a test statistic greater than 1 .886.

02:27 So i just need to go to the normal cdf.

02:30 There are different ways you can do this, but normal cdf, i think this is the easiest way.

02:34 And our lower bound is going to be our test statistics, so 1 .886.

02:40 The upper bound is any large number.

02:41 I'm going to go e99.

02:43 That's the biggest number this calculator can handle, but you can put whatever large number you want...

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