Consider the following hypothesis test: H0: μ=100 H2: μ+100 A sample of 65 is used.

Consider the following hypothesis test: H0: μ=100 ...

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical framework used to make decisions about a population parameter based on sample data. It involves setting up a null hypothesis that represents a status quo or a baseline assumption, and an alternative hypothesis that represents a competing claim. The goal is to use evidence from the sample to determine whether the null hypothesis can be rejected in favor of the alternative hypothesis.

Null and Alternative Hypotheses

The null hypothesis (H?) represents the claim to be tested, typically stating that there is no effect or no difference (for example, that a mean equals a specific value). The alternative hypothesis (H?) represents the claim that contradicts the null, indicating a significant difference or effect. In many practical cases, the alternative is two-sided, meaning the parameter can be either less than or greater than the value stated in the null hypothesis.

t-Test for the Mean

A t-test is a statistical test used to evaluate hypotheses about a population mean when the population standard deviation is unknown and the sample size is relatively small or moderate. It involves using the sample mean, the sample standard deviation, and the sample size to estimate how far the observed mean is from the hypothesized mean under the null hypothesis. The t-test is especially useful when the data are assumed to be approximately normally distributed.

Test Statistic Calculation

The test statistic in a t-test is calculated by taking the difference between the sample mean and the hypothesized population mean, and then dividing that difference by the standard error of the sample mean. The standard error is computed by dividing the sample standard deviation by the square root of the sample size. This statistic follows a t-distribution with degrees of freedom equal to the sample size minus one.

p-Value Interpretation

The p-value represents the probability of obtaining test results at least as extreme as those observed during the test, assuming the null hypothesis is true. A small p-value indicates that the observed data would be unlikely under the null hypothesis, thus providing evidence against it. In hypothesis testing, the p-value is used as a measure of the strength of evidence against the null hypothesis.

Significance Level and Decision Making

The significance level (commonly denoted by ?) is the threshold probability used to decide whether to reject the null hypothesis. If the p-value is less than or equal to ?, the null hypothesis is rejected, suggesting that the observed effect is statistically significant. Conversely, if the p-value exceeds ?, there is insufficient evidence to reject the null hypothesis. This decision-making process is central to statistical inference in hypothesis testing.

Transcript

00:01 Hello, so here we're giving the sample size n is 65, so the degrees of freedom is then 64.

00:05 The sample mean x bar is 103, and the sample standard deviation is 11 .5.

00:11 So the hypotheses here to be tested would be h0, where we have mu is going to be equal to 100.

00:19 And then the alternative hypothesis would be that mu would not be equal to 100.

00:26 So here the test statistic for the hypothesis test about a population where sigma is known is given by t is equal to x bar minus mu not.

00:37 That's going to be 103 minus 100 and then divided by s divided by square root of n.

00:45 So 11 .5 divided by the square root of 64, which gives us 2 .09.

00:53 So then we get here that the area under the t distribution curve to the left of negative 2 .09 is the same as the area under the curve to the right of t equals 2 .09, which we obtain from the t table.

01:08 And we see that t equals 2 .098 is between 1 .998 and 2 .386.

01:13 We then see the area in the tail on the right of t equals 2 .09 is between 2 .025 and 0 .01.

01:21 We then double these amounts and get the p value then must be between 0 .05 and 0 .02.

01:32 So we have here that our p value then is going to be less than alpha, where alpha is 0 .05.

01:40 So therefore, we reject the null hypothesis at a 5 % level of significance and then conclude that mu is not equal.

01:52 Equal to 100.

01:54 So we can reject the null hypothesis and then we can accept the alternative hypothesis.

01:59 And then for part b, we are given that our sample size n is 65, degrees of freedom 64, sample mean 96 .5, and a standard deviation is 11.

02:11 So now we get that our t statistic is going to be equal to x bar minus mu not, which is going to be 96 .5 minus 100...

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