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

Large Sample and Central Limit Theorem

When samples are sufficiently large, the Central Limit Theorem (CLT) asserts that the sampling distribution of the test statistic approximates a normal distribution, regardless of the original data's distribution. This property allows statisticians to use z-tests or other large-sample approximations to perform hypothesis testing effectively, even when the underlying population distribution is unknown.

Critical Value Approach

The critical value approach in hypothesis testing involves comparing a test statistic—computed from sample data—to a threshold value known as the critical value, which is determined by the chosen significance level of the test. If the test statistic exceeds this critical value (or falls into the critical region), then the null hypothesis is rejected. This method provides a clear decision rule based on predetermined criteria.

p-value

The p-value is the probability, assuming that the null hypothesis is true, of obtaining a test statistic as extreme as or more extreme than the value actually observed. It quantifies the evidence against the null hypothesis; a smaller p-value indicates stronger evidence in favor of the alternative hypothesis. In hypothesis testing, the p-value is compared to the significance level to decide whether to reject the null hypothesis.

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, or hypothesis, about a population is true. It involves stating a null hypothesis (typically a statement of no effect or status quo) and an alternative hypothesis, then using sample data to determine which hypothesis is more consistent with the observed results.

Null and Alternative Hypotheses

The null hypothesis (H?) and the alternative hypothesis (H?) form the basis of any statistical test. The null hypothesis usually represents a default or status quo assumption while the alternative represents the claim being tested. The process involves assessing whether the observed sample data significantly deviates from the expectation under the null hypothesis to warrant rejecting it in favor of the alternative.

Transcript

00:01 The following is the solution to number seven, and we have a hypothesis test here, where p1 minus p2 is 0 .25, or the alternative being p1 minus p2 is less than 0 .25.

00:09 And we're testing this at the 0 .5 % level of significance given this data.

00:14 So what we're going to do is we're going to use this formula up here to find the test statistics.

00:19 So z is equal to 0 .57 minus 0 .37, which is the observed differences minus the hypothesized difference of 0 .25.

00:30 And then that's divided by this standard error of the estimate.

00:36 So it's p1 hat, so 0 .57 times 1 minus p1 hat, which would be 0 .43 over that sample size of 1 ,400, plus p2 hat, which is 0 .37, times 1 minus p2 hat, which is 0 .63.

00:55 Divided by that other sample size of 1 ,200, and that should give us negative 2 .602.

01:02 So a smaller test statistic there.

01:06 Now we need to find z star or the critical value.

01:09 It's going to tell us whether or not to reject or not reject.

01:11 So this is a one -tail test, a left -tail test actually, with .05 as our alpha.

01:18 So if we type in .005 and then we change this to left, that's going to give us our test statistic of negative 2 .576.

01:30 Okay, so negative 2 .576 is our critical value and anything to the left of that we would reject since it's a left tail test and this is slightly to the left of it so it is in the rejection zone.

01:46 So i'm going to reject the h0.

01:49 Now to find the p value because i'm also supposed to find the p value, i find the probability that i get a test statistic less than negative 2 .602.

01:58 So if i need to find the probability, i'm going to go to normal cds, since this is a normal distribution.

02:03 The lower bound is negative infinity, and the upper bound is going to be that negative to .602.

02:08 And the mean and standard deviation are 01 always.

02:12 So that's .0046 as the p value.

02:16 So 0 .0046.

02:20 So you can see that's a nice double check.

02:22 That is slightly less than that .005, so i do in fact reject.

02:27 So the next one's similar.

02:28 It's p1 minus p2 is .16, and then p1 minus p2 is not equal to.

02:31 So this is a two -tail test where the alpha equals .02, given this data.

02:35 So i still find my z value...