An article in the San Jose Mercury News stated that students...

An article in the San Jose Mercury News stated that students in the California state university system take 4.5 years, on average, to finish their undergraduate degrees. Suppose you believe that the mean time is longer. You conduct a survey of 49 students and obtain a sample mean of 5.1 with a sample standard deviation of $1.2 .$ Do the data support your claim at the 1\% level?

Key Concepts

Test Statistic

The test statistic is a standardized value calculated from sample data that measures the degree of deviation of the sample statistic from the hypothesized population parameter under the null hypothesis. In the context of a t-test, it is computed by taking the difference between the sample mean and the hypothesized mean, divided by the standard error.

Significance Level

The significance level, denoted by alpha (?), is the threshold set by the researcher for rejecting the null hypothesis. It quantifies the risk of making a Type I error, which is rejecting the null hypothesis when it is true, with lower values requiring stronger evidence to justify rejection.

p-Value

The p-value is the probability of observing data as extreme as, or more extreme than, what was actually observed under the assumption that the null hypothesis is correct. A small p-value indicates that such extreme outcomes are unlikely under the null hypothesis, providing evidence in favor of the alternative hypothesis.

Null and Alternative Hypotheses

The null hypothesis (H0) is the default assumption about a population parameter and typically represents a statement of no effect or no difference, while the alternative hypothesis (Ha) represents the statement that the researcher aims to support, indicating a change, effect, or difference relative to the null hypothesis.

Hypothesis Testing

Hypothesis testing is a methodological framework used to determine whether sample data provide enough evidence to reject a presumed claim about a population parameter. It involves comparing the observed data to what is expected under a specific hypothesis, often called the null hypothesis, to evaluate the plausibility of that hypothesis.

t-Test for the Mean

The t-test is a statistical procedure used to compare a sample mean with a hypothesized population mean when the population standard deviation is unknown. It uses the t-distribution to account for sampling variability, especially in small samples, and helps determine whether the difference between the sample mean and the hypothesized mean is statistically significant.

Transcript

00:01 Welcome to this lesson.

00:02 In this lesson we have an article that states that the california state university system takes 4 .5 years on average to finish an undergraduate degree.

00:14 So we conducted a survey taking 49 students and we obtained a sample mean of 5 .1 for their years of completion and a standard deviation of 1 .2 years.

00:32 So we are looking at or we are testing whether or not our data supports that claim.

00:38 So the first part, the first step to this is to have the statistical hypothesis where we state a null and alternative hypothesis.

00:53 So we have the null hypothesis that sees that the mean is equals to 4 .5.

01:03 Alright and we would want to test our claim so we have the mean which is greater than 4 .5.

01:13 Alright so this has become an aperture test.

01:18 The step 2 would identify a level of significance.

01:22 So here we have the alpha which is 0 .01 and as such we need to find the critical value so probability that or the z -score for this probability given that we have 0 .19 we have 0 .99 okay of the area to the left so we have the critical value z critical as and that's equal to 2 .33.

01:59 Alright so let's go to the step 3 where we look at our test statistic.

02:15 So here we have the z -score that's equals to the x value or the sample mean minus the hypothesized mean divided by the standard deviation on the square root of the sample size...

Please give Ace some feedback