Test the hypotheses H0: μ=100 H1: μ<100 using a random sample of n=36, a probability

step 1 Hello, so here we have a sample size and is equal to 36. The probability of a type 1 error, alph

Test the hypotheses H0: μ=100 H1: μ<100 using a...

Key Concepts

Decision Rule

The decision rule in hypothesis testing specifies that if the calculated test statistic falls into the rejection region (or if the p-value is below the predetermined significance level), the null hypothesis is rejected in favor of the alternative hypothesis. Otherwise, there is insufficient evidence to reject the null hypothesis.

p-value

The p-value is the probability of obtaining a sample statistic as extreme as the one observed, assuming that the null hypothesis is true. It quantifies the evidence against the null hypothesis; a smaller p-value indicates stronger evidence for rejecting the null hypothesis when it falls below the significance level.

One-Tailed Test

A one-tailed test is used when the research hypothesis specifies a direction of the effect (for example, testing whether the mean is less than a certain value). In this type of test, all of the significance level (?) is placed in one tail of the distribution, which determines the critical region for rejecting the null hypothesis.

Hypothesis Testing

Hypothesis testing is a framework in statistics for making decisions about a population parameter based on sample data. It involves setting up a null hypothesis, which is a default assumption regarding the parameter, and an alternative hypothesis that reflects what the researcher wants to test against the null hypothesis.

Type I Error

Type I error is the probability of rejecting the null hypothesis when it is actually true. This error is controlled by the significance level (?), which is predetermined and represents the maximum acceptable probability of making a Type I error in the hypothesis test.

Null and Alternative Hypotheses

The null hypothesis (H0) is a statement asserting that there is no effect or no difference, often specifying a particular value for a parameter (such as a mean). In contrast, the alternative hypothesis (H1) states that there is an effect or a difference, and it can be directional (one-tailed) or non-directional (two-tailed) depending on the nature of the test.

Test Statistic

The test statistic is a numerical summary computed from the sample data which, under the null hypothesis, follows a known probability distribution. When the population variance is unknown and the sample size is moderate, the t-test is typically used, calculating the test statistic by comparing the difference between the sample mean and the hypothesized mean relative to the estimated standard error.

Transcript

00:01 Hello, so here we have a sample size and is equal to 36.

00:03 The probability of a type 1 error, alpha, is equal to 0 .05, and then our null hypothesis, mu is equal to 100 versus our alternative hypothesis that mu is going to be less than 100.

00:15 Now, our population mean is mu not equal to 100, so the alternative hypothesis is left -tailed.

00:22 From the given information, the population variance is unknown.

00:25 So we apply a one sample t test.

00:28 And then in part a here, we have our sample mean x bar is going to be equal to 106.

00:36 Sample standard deviation is equal to 15.

00:40 The degrees of freedom is n minus 1.

00:43 So we have 35 degrees of freedom.

00:48 And then using the excel function for the critical value, we get our t critical value being equal to negative 1 .6896.

00:57 So our decision is to reject the null hypothesis if t is less than negative 1 .6896.

01:06 So t here is equal to x bar minus mu not divided by s over the square root of n.

01:11 So we get that our test statistic is going to be equal to 106 minus 100 divided by 15 divided by 6 or the square root of 36, which is equal to 2 .4, which we observe is going to be greater.

01:28 Then negative 1 .6896.

01:30 So therefore we fail to reject the null hypothesis.

01:39 Therefore, it can be concluded that the population mean is not less than 100.

01:45 And then in part b, we have the sample mean x bar is equal to 104.

01:52 Standard deviation, s is equal to 10.

01:55 Again, 35 degrees of freedom now.

01:57 So we can again use the excel function to find the t critical value, which is going to be given at 0 .05 with 35 degrees of freedom, giving us negative 1 .6896.

02:14 And then our decision is to reject h -0 if t is less than negative 1 .6896...

Please give Ace some feedback