The null and alternate hypotheses are: H0: μd≤ 0 H1: μd>0 The following sample informa

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

The null and alternate hypotheses are: $$\begin{array}{l}H_{0}: \mu_{d} \leq 0 \\H_{1}: \mu_{d}>0\end{array}$$The following sample information shows the number of defective units produced on the day shift and the afternoon shift for a sample of four days last month. At the .05 significance level, can we conclude there are more defects produced on the afternoon shift?

The null and alternate hypotheses are:
$$\begin{array}{l}H_{0}: \mu_{d} \leq 0 \\H_{1}: \mu_{d}>0\end{array}$$The following sample information shows the number of defective units produced on the day shift and the afternoon shift for a sample of four days last month. At the .05 significance level, can we conclude there are more defects produced on the afternoon shift?

Key Concepts

One-Tailed Test

A one-tailed test is a type of hypothesis test where the alternative hypothesis specifies a direction of the effect, focusing on only one tail of the distribution. This approach is used when the research hypothesis predicts not just a difference, but a difference in a specific direction.

Type I Error

A Type I error occurs when the statistical test incorrectly rejects the null hypothesis when it is actually true. The significance level defines the likelihood of making this error, underscoring the importance of setting an appropriate threshold for decision-making.

Significance Level

The significance level is a threshold set prior to conducting a test, indicating the maximum acceptable probability of committing a Type I error, which is rejecting a true null hypothesis. It is commonly set at 0.05 in many studies.

Null Hypothesis

The null hypothesis is a formal statement asserting that there is no effect or no difference in the parameter of interest. In the context of comparing means, it typically states that the difference between conditions or treatments is less than or equal to zero, reflecting no increase or improvement.

Alternative Hypothesis

The alternative hypothesis is a statement that contradicts the null hypothesis and is what a researcher aims to support with evidence. It posits that there is an effect or difference—in this case, that the mean difference exceeds zero—indicating a directional change in one of the compared conditions.

Paired Sample t-Test

A paired sample t-test is a statistical method used when comparing two related or matched samples. It accounts for the dependence between paired observations by analyzing the differences within each pair, thereby increasing the test's sensitivity when detecting a mean difference.

Transcript

00:01 So we have data for the day shift and the afternoon shift, and the day shift a number of defects or problems are listed here, and then the 8, 9, 12, and 15.

00:15 And we're supposed to find out, do we have evidence that the afternoon shift has more defects than the day and using a 5 % significance level? and it seems kind of like a silly question because all the afternoons are smaller numbers.

00:36 So the answer is going to be no.

00:40 What will carry out the hypothesis test? i suspect that they may have written it the wrong way.

00:45 But we're going to have the difference be the afternoon score less the day score.

00:53 And so we have our mean listed.

00:59 As less than or equal to zero as our assumption and alternately that the difference is greater than zero.

01:09 And so our differences, and i'll put those in blue, that difference is negative two, that difference is negative three, that difference is negative three, and that difference is negative four.

01:20 So when i go through and find the mean difference for those values, and let me clear that out, we find that the mean difference is negative 3 for that sample, and i find that the standard deviation of those differences comes out as .816, and it would round to 5, and our sample size is only 4.

01:45 And so we would have to use a test statistic that is a t value, and it would only have three degrees of freedom.

01:52 And if we wanted to find our critical value, a critical value would have for our 5 % 3 % percent.

02:00 Significance level would have all 5 % in that upper tail since this is greater than.

02:06 And again, as i'm saying, this test doesn't really make sense.

02:09 And the value that has 5 % in the upper tail is a t value, that critical value, 2 .353.

02:19 And we would be rejecting our null if we were having our value higher, which we're actually not going to because we have this as a negative difference...

Please give Ace some feedback