Test the hypotheses H0: μ≤100 H1: μ>100 using a random sample of n=25, a probability

Test the hypotheses H0: μ≤100 H1: μ>100 using a...

Key Concepts

Degrees of Freedom

Degrees of freedom, in the context of a t-test, typically refer to the number of independent pieces of information available for estimating a parameter. For a one-sample t-test, it is calculated as the sample size minus one, which influences the shape of the t-distribution used in the test.

Test Statistic

The test statistic summarizes the difference between the sample data and the null hypothesis in a standardized form. In a t-test, it is typically calculated as the sample mean minus the hypothesized mean, divided by the estimated standard error of the mean.

Rejection Region

The rejection region is the range of values for the test statistic that leads to the rejection of the null hypothesis. It is determined by the significance level and the nature of the test (one-tailed or two-tailed), and in a one-tailed test, it lies entirely in one tail of the probability distribution.

t-Test

The t-test is a statistical test used when the population standard deviation is unknown and the sample size is relatively small. It assesses whether the sample mean significantly differs from a hypothesized population mean, and it uses the t-distribution to determine critical values.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves setting up two competing hypotheses, the null (H0) and the alternative (H1), and then using a test statistic calculated from the sample data to decide whether to reject the null hypothesis in favor of the alternative.

Type I Error

A Type I error occurs when the null hypothesis is incorrectly rejected when it is actually true. The probability of committing a Type I error is represented by the significance level (?), and controlling this error is crucial in hypothesis testing.

Alternative Hypothesis

The alternative hypothesis (H1) is the statement that contradicts the null hypothesis, suggesting that the parameter takes on a different value than stated in H0. In the context of one-sided tests, it specifies a direction of the effect, such as the mean being greater than the stated value.

Null Hypothesis

The null hypothesis (H0) represents a statement of no effect or status quo, and in many tests it is the claim that the parameter (such as a mean) is less than or equal to, or equal to, a specific value. It serves as a baseline statement that the statistical test seeks to challenge with evidence from sample data.

Significance Level

The significance level, often denoted by ?, is the threshold for determining whether a test result is statistically significant. It represents the maximum probability of making a Type I error that one is willing to accept, such as 0.05 in many tests.

Transcript

00:01 Hello, so here we're going to test the null hypothesis, h0, which is going to be mu less than equal to 100, against the alternative hypothesis, h1, which is going to be mu greater than 100, using a random sample of n is equal to 25 and the probability of a type 1 error, or alpha is equal to 0 .05.

00:20 So then we have the values here in part a are going to be x bar, which is going to be equal to 106, and then we have s is equal to 15.

00:29 So then the value of the population mean u not equals 100 is going to be tested by our test statistic here, which is going to be given by 106 minus 100 divided by 15 divided by the square root of 25, which is going to be equal to 2.

00:48 So the critical value of our test statistic is then 1 .711, which the calculated value for t is more, then the calculated value of 1 .711.

01:04 So here we can reject the null hypothesis.

01:08 So we reject the null hypothesis with a 5 % level of significance.

01:18 And then we accept the alternative hypothesis that the value of the population mean is going to be more than 100.

01:27 Then in part b, the values are x bar, which is going to be equal to 104.

01:33 And now s is equal to 10.

01:35 So then our calculated value for t is 104 minus 100 again over 10 over square 25, which again is equal to 2.

01:44 And then our critical value is 1 .711.

01:50 So here the calculated value for t is more than our calculated value.

01:56 So we reject the null hypothesis again with a 5 % level of significance...

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