The null and alternate hypotheses are: H0: μd=0 H1: μd≠ 0 The following paired observ

The null and alternate hypotheses are: H0: μd=0 ...

The null and alternate hypotheses are: $$\begin{array}{l}H_{0}: \mu_{d}=0 \\H_{1}: \mu_{d} \neq 0\end{array}$$ The following paired observations show the number of traffic citations given for speeding by Officer Dhondt and Officer Meredith of the South Carolina Highway Patrol for the last five months. At the .05 significance level, is there a difference in the mean number of citations given by the two officers?

The null and alternate hypotheses are:
$$\begin{array}{l}H_{0}: \mu_{d}=0 \\H_{1}: \mu_{d} \neq 0\end{array}$$
The following paired observations show the number of traffic citations given for speeding by Officer Dhondt and Officer Meredith of the South Carolina Highway Patrol for the last five months. At the .05 significance level, is there a difference in the mean number of citations given by the two officers?

Key Concepts

Null Hypothesis

The null hypothesis is a statement asserting that there is no effect or no difference between the groups or conditions being compared. In statistical testing, it functions as a baseline from which any deviation is considered evidence against it. This concept is crucial because it forms the foundation of hypothesis testing, allowing researchers to determine if observed differences are statistically significant or likely due to random chance.

Alternative Hypothesis

The alternative hypothesis represents the claim that there is a true effect or a difference between the groups or conditions being compared. It is what the researcher aims to support through the gathered data. This hypothesis is essential in statistical testing, as its acceptance or rejection (in favor of the null hypothesis) is determined by assessing the evidence provided by the sample data.

Paired Sample t-Test

The paired sample t-test is a statistical procedure used to compare the means of two related groups or matched pairs. This test is particularly applicable when the same subjects are measured under two different conditions or when pairs of subjects are matched based on certain characteristics. By accounting for the natural pairing in the data, this test more accurately assesses differences between the paired observations, reducing variability and increasing test sensitivity.

Significance Level

The significance level is a threshold used in hypothesis testing to determine whether to reject the null hypothesis. Commonly denoted by alpha (?), it represents the maximum probability of making a Type I error—rejecting a true null hypothesis. A typical value like 0.05 indicates that there is a 5% risk of concluding that a difference exists when there is none. This concept helps ensure that the findings are not merely due to chance.

Transcript

00:01 So we're looking at d being the difference between, and i'll call them officer d minus a.

00:06 And we're to assume that the number that mean difference in citations, and we're subtracting them in this way because for each one of these, it's for the same month period.

00:17 So we're going to assume that that difference is zero and alternately that it is not zero.

00:23 So that they want to find out is there a significant difference between the citations they give for month? and my four differences when i subtracted this way were 4, 3, 5, 4, and 7.

00:37 So that's our data.

00:38 So we only have five numbers for our differences.

00:42 And the mean of those differences comes out to be, and let me quick put this back in, the mean of those differences comes out to be 4 .6.

00:55 And the sample standard deviation for those five numbers is 1 .410.

01:02 Six and i'm going to call it six.

01:04 And so we want to find at a 5 % significance level, do we have evidence to show that there's a difference between the number of citations? and we're not doing a direction.

01:16 So we can do this one of two ways.

01:18 We can either go through and find with our degrees of freedom, degrees of freedom being equivalent to four.

01:26 We can find if we want 5 % significance, we can find what those t -based, values are that are the critical values for our setting...

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