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

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? Note: Use the six-step hypothesis testing procedure to solve the following exercises.

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? Note: Use the six-step hypothesis testing procedure to solve the following exercises.

Key Concepts

Paired T-Test

A paired t-test is a statistical technique used when comparing two related samples, matched samples, or repeated measurements on a single sample in order to determine whether their means differ significantly. It is particularly useful when the observations are paired or matched in some way, thereby controlling for variability between subjects.

Significance Level

The significance level, often denoted by alpha (?), is the threshold for statistical significance in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true (a Type I error). A commonly used significance level is 0.05, indicating a 5% risk of this error.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves comparing an observed test statistic to a theoretical distribution, which allows researchers to quantify the strength of evidence against a null hypothesis.

Null and Alternative Hypotheses

The null hypothesis (H?) represents a default assumption about a population parameter, typically stating that there is no effect or no difference, while the alternative hypothesis (H?) represents a competing claim that indicates the presence of an effect or difference. These hypotheses are the foundation for making inferences in hypothesis testing.

Six-Step Hypothesis Testing Procedure

The six-step hypothesis testing procedure is a systematic approach to conducting a hypothesis test. The steps include: defining the null and alternative hypotheses, choosing the significance level, selecting the appropriate test statistic, calculating the test statistic using sample data, determining the critical value or p-value, and finally making a decision to either reject or fail to reject the null hypothesis based on the comparison of the test statistic with the critical value or p-value with the significance level.

Transcript

00:01 So we will be assuming that the difference we're going to have is officer d minus m.

00:07 That's the way i'm going to subtract.

00:08 We could go the other way, but we'll just do it in that order.

00:11 We're assuming that that difference is zero, and alternately, that the difference is not zero.

00:17 So we want to show a two -tail test in this case.

00:20 And i have the differences as four, three, five, four, and seven.

00:26 So we only have a sample size of five, not a lot.

00:30 And when we do our calculation for this particular test, and we get that the mean and standard deviation of that data, the mean is 4 .6 citations, and the standard deviation is 1 .516, and i'll call it 6.

00:53 And so we will be assuming we're using a 5 % significance level.

01:00 And so let's draw a little picture here.

01:03 We're assuming that the mean difference is equal to zero for that sampling distribution of sample sizes of five.

01:11 And since we're doing a two -tail test, we're going to have our values with the 0 .025 in each tail.

01:21 And this is where we would reject the null down here and also down here with that tail being the lower tail being 0 .025.

01:30 And we would reject the null down here.

01:32 And so we need to find the t value with four degrees of freedom that has that 2 .5 % in the upper tail.

01:39 And that value is 2 .776.

01:44 So anything higher, greater than or equal to that, we would reject the null and anything less than are equal to negative 2 .76...

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