Examples: For each of the following examples, we will first determine which kind of problem it is

Review of Statistical Inference from STA 2023

Confidence Intervals and Significance Tests for:

· one mean

· matched paired differences

· difference of two independent means

· one proportion

· difference of two independent proportions

Chapter 7: CI and Significance Tests for means using the t distribution

1. Why do we use the t table?

2. What assumptions do we need to make? How do we check them?

· Random Samples

· Population is Normal

3. Find Formulas on tables.

Confidence Interval:

Significance Test: H_o: H_a:

TS:

p-value:

Read Minitab Output

Interpret Results

· CI does NOT include # from H_o → Supports H_a

· p-value small → Supports H_a

Examples: For each of the following examples, we will first determine which kind of problem it is. Then we will set the problem up as if to do by hand. We will interpret the Minitab output given for each problem.

1. Do pregnant women who use cocaine have babies with lower birth weight than women who do not use cocaine? Pregnant women were tested for cocaine/crack, and the birth weights of babies (in grams) were recorded and averaged for women who tested positive, and those who tested negative separately.

	n		s
Positive Test	134	2733	599
Other	5974	3118	672

2. Many children are diagnosed each year with asthma. In an effort to educate these children about their condition, an educational video was developed. To test the effectiveness of this video, ten randomly selected children, of elementary school age, who had been recently diagnosed, were chosen to participate in a study. A nurse asked the children a series of questions about asthma, then showed them the video and asked the questions again. The children’s scores follow:

Child	1	2	3	4	5	6	7	8	9	10
Before	61	60	52	74	64	75	42	63	53	56
After	67	62	54	83	60	89	44	67	62	57

3. A news report states that over 70% of mail received by a household in a week is advertisements. A sample of 20 households produced the following data:

adv mail %adv

12 16 0.750000

18 24 0.750000

15 24 0.625000

15 25 0.600000

21 33 0.636364

21 24 0.875000

17 22 0.772727

20 27 0.740741

16 28 0.571429

13 19 0.684211

20 25 0.800000

20 23 0.869565

17 20 0.850000

13 20 0.650000

23 33 0.696970

15 28 0.535714

9 12 0.750000

18 30 0.600000

9 14 0.642857

24 35 0.685714

Two-Sample T-Test and CI

Sample N Mean StDev SE Mean

1 134 2733 599 52

2 5974 3118 672 8.7

Difference = mu (1) - mu (2)

Estimate for difference: -385.000

95% CI for difference: (-488.738, -281.262)

T-Test of difference = 0 (vs <): T-Value = -7.34 P-Value = 0.000 DF = 140

Paired T-Test and CI: before, after

Paired T for before - after

N Mean StDev SE Mean

before 10 60.0000 10.0000 3.1623

after 10 64.5000 13.2267 4.1826

Difference 10 -4.50000 5.12619 1.62104

95% CI for mean difference: (-8.16705, -0.83295)

T-Test of mean difference = 0 (vs < 0): T-Value = -2.78 P-Value = 0.011

One-Sample T: adv

Variable N Mean StDev SE Mean 95% CI

adv 20 16.8000 4.2501 0.9503 (14.8109, 18.7891)

One-Sample T: mail

Variable N Mean StDev SE Mean 95% CI

mail 20 24.1000 6.2061 1.3877 (21.1955, 27.0045)

One-Sample T: %adv

Variable N Mean StDev SE Mean 95% CI

%adv 20 0.704315 0.098776 0.022087 (0.658086, 0.750543)

Variable N Mean StDev SE Mean T P

%adv 20 0.704315 0.098776 0.022087 0.20 0.424

Positive Test

Before