Inferences About Two Means: Dependent Samples

Inferences About Two Means: Dependent Samples • • Definition Two samples are independent if the sample values from one population are not related to or somehow naturally paired with the sample values from the other population.

Two samples are dependent if the sample values are paired.

Inferences About Two Means: Dependent Samples Notation: 𝑑 = individual difference between the two values in a single matched pair.

𝑢 𝑑 = mean value of the difference population of all pairs of data.

d for the 𝑑 =mean value the differences d sample data.

for the paired 𝑠 𝑑 = Standard deviation of the difference paired sample data.

d for the 𝑛 = number of pairs of sample data.

Inferences About Two Means: Independent Samples Requirements: 1. The two samples are Dependent.

2. Both samples are simple random samples.

3. Both populations have sample size 𝑛 > 30 or both samples are normal.

Inferences About Two Means: Dependent Samples Hypothesis Test Statistic • with df = 𝑛 − 1 𝑡 = 𝑑 − 𝜇 𝑑 𝑠 𝑑 𝑛

Inferences About Two Means: Dependent Samples Hypothesis Test 1. Identify claim and state Hypotheses. (𝐻 0 : 𝜇 𝑑 𝐻 𝑎 is that 𝜇 𝑑 is different in some way).

2. Find the Test Statistic (here we find a 𝑡 stat) = 0, In calc 𝐿 1 − 𝐿 2 → 𝐿 3 then Ttest on 𝐿 3 3. Draw a picture and find the p-value.

4. Make a decision (compare p-value and 𝛼) 5. Summarize

Inferences About Two Means: Dependent Samples The data below is a small sample of measure weights of college students in September and April of their freshman year. Using a 0.05 significance level test the claim that for the population of students, the mean change in weight from September to April is equal to 0 kg.

April Weight September Weight Difference 66 67 -1 52 53 -1 68 64 4 69 71 -2 71 70 1

Inferences About Two Means: Dependent Samples Researchers collected data on the numbers of hospital admissions resulting from motor vehicle crashes, and results are given below for Fridays on the 6 th of the month and Fridays on the following 13 th of the month. Use a 0.05 significance level to test the claim that when the 13 th day of a month falls on a Friday, the numbers of hospital admissions from motor vehicle crashes are not affected.

Friday the 6 th Friday the 13 th 9 13 6 12 11 14 11 10 3 4 5 12

Inferences About Two Means: Dependent Samples Homework: 9-3: 9-21 odd