In: Statistics and Probability
Six sets of identical twins were randomly selected from a population of identical twins. One child was taken at random from each pair to form an experimental group. These children participated in a program designed to promote creative thinking. The other child from each pair was part of the control group that did not participate in the program to promote creative thinking. At the end of the program, a creative problem-solving test was given, with the results shown in the table below.
Twin Pair | A | B | C | D | E | F |
Experimental group | 53 | 33 | 12 | 25 | 33 | 47 |
Control Group | 37 | 23 | 7 | 18 | 21 | 42 |
Higher scores indicate better performance in creative problem
solving. Do the data support the claim that the program of the
experimental group did promote creative problem solving? Use
α = 0.01. (Let d = experimental − control.)
What are we testing in this problem?
single proportionpaired difference difference of proportionsdifference of meanssingle mean
What is the level of significance?
State the null and alternate hypotheses.
H0: μd = 0; H1: μd ≠ 0H0: μd > 0; H1: μd ≤ 0 H0: μd ≤ 0; H1: μd > 0H0: μd ≠ 0; H1: μd = 0
What sampling distribution will you use? What assumptions are you
making?
The Student's t. We assume that d has an approximately uniform distribution.The standard normal. We assume that d has an approximately normal distribution. The Student's t. We assume that d has an approximately normal distribution.The standard normal. We assume that d has an approximately uniform distribution.
What is the value of the sample test statistic? (Round your answer
to three decimal places.)
Estimate the P-value.
P-value > 0.2500.125 < P-value < 0.250 0.050 < P-value < 0.1250.025 < P-value < 0.0500.005 < P-value < 0.025P-value < 0.005
Sketch the sampling distribution and show the area corresponding to
the P-value.
Will you reject or fail to reject the null hypothesis? Are the data
statistically significant at level α?
At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
Interpret your conclusion in the context of the application.
There is sufficient evidence at the 0.01 to conclude that the program of the experimental group promoted creative problem solving.There is insufficient evidence at the 0.01 to conclude that the program of the experimental group promoted creative problem solving.
paired difference
level of significance =0.01
H0: μd ≤ 0; H1: μd > 0
The Student's t. We assume that d has an approximately normal distribution.
S. No | experimental | Control | diff:(d)=x1-x2 | d2 |
1 | 53 | 37 | 16 | 256.00 |
2 | 33 | 23 | 10 | 100.00 |
3 | 12 | 7 | 5 | 25.00 |
4 | 25 | 18 | 7 | 49.00 |
5 | 33 | 21 | 12 | 144.00 |
6 | 47 | 42 | 5 | 25.00 |
total | = | Σd=55 | Σd2=599 | |
mean dbar= | d̅ = | 9.167 | ||
degree of freedom =n-1 = | 5.000 | |||
Std deviaiton SD=√(Σd2-(Σd)2/n)/(n-1) = | 4.36 | |||
std error=Se=SD/√n= | 1.7780 | |||
test statistic = | (d̅-μd)/Se = | 5.156 |
P-value < 0.005
At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.
There is sufficient evidence at the 0.01 to conclude that the program of the experimental group promoted creative problem solving