Remember that a complete answer to a hypothesis test question includes 4 steps:
ID: 3228907 • Letter: R
Question
Remember that a complete answer to a hypothesis test question includes 4 steps:
1) Hypothesize: Write a null and alternative hypothesis
2) Prepare: Make sure your sample size is large enough to satisfy the CLT for sample proportions (Remember to use the POOLED SAMPLE PROPORTION and verify that BOTH SAMPLE SIZES are large enough.)
3) Test Statistic: Computed using technology (StatCrunch or a TI Calculator)
4) Interpret: Write a conclusion in PLAIN ENGLISH related to the given problem
1. Researchers were investigating the effectiveness of nicotine gum in helping people quit smoking. Of 1649 people using nicotine gum, 174 succeeded in quitting smoking. Of 1648 people receiving a placebo, 66 succeeded in quitting smoking. Do we have evidence that nicotine gum is effective in helping people quit smoking? Remember to show all 4 steps of your hypothesis test.
2. In a study, people were observed for about 10 seconds in public places to determine whether they smiled during the randomly chosen 10-second interval. The table shows the results comparing males and females. Male with Smile 3269, without smile 3806. Female with smile 4471, without smile 4278.
a. Do we have evidence that there is a difference in the proportion of males and females who smile? Remember to show all 4 steps of your hypothesis test.
b. Construct a 95% confidence interval for the difference in the proportion of males and females who smile. How does your confidence interval support your conclusion in part a?
Explanation / Answer
Question 1
Here, we have to use the two sample z test for the population proportions.
The null and alternative hypothesis for this test is given as below:
Null hypothesis: H0: There is no any statistically significant difference in the population proportions of quitting smoking by using nicotine gum and placebo.
Alternative hypothesis: Ha: There is a statistically significant difference in the population proportions of quitting smoking by using nicotine gum and placebo.
Test statistic formula is given as below:
Z = (P1 – P2) / sqrt[(p1q1/n1)+(p2q2/n2]
The test is given as below:
Data
Hypothesized Difference
0
Level of Significance
0.05
Group 1
Number of Items of Interest
174
Sample Size
1649
Group 2
Number of Items of Interest
66
Sample Size
1648
Intermediate Calculations
Group 1 Proportion
0.105518496
Group 2 Proportion
0.040048544
Difference in Two Proportions
0.065469952
Average Proportion
0.0728
Z Test Statistic
7.2350
Two-Tail Test
Lower Critical Value
-1.9600
Upper Critical Value
1.9600
p-Value
0.0000
Reject the null hypothesis
Question 2
Part a
Here, we have to use the two sample z test for the population proportions.
The null and alternative hypothesis for this test is given as below:
H0: There is no any significant difference in the population proportions of male with smile and female with smile.
Ha: There is a significant difference in the population proportions of male with smile and female with smile.
Test statistic formula is given as below:
Z = (P1 – P2) / sqrt[(p1q1/n1)+(p2q2/n2]
We assume 5% level of significance for this test.
The test is given as below:
Z Test for Differences in Two Proportions
Data
Hypothesized Difference
0
Level of Significance
0.05
Male
Number of Items of Interest
3269
Sample Size
7075
Female
Number of Items of Interest
4471
Sample Size
8749
Intermediate Calculations
Group 1 Proportion
0.46204947
Group 2 Proportion
0.511029832
Difference in Two Proportions
-0.04898036
Average Proportion
0.4891
Z Test Statistic
-6.1283
Two-Tail Test
Lower Critical Value
-1.9600
Upper Critical Value
1.9600
p-Value
0.0000
Reject the null hypothesis
We reject the null hypothesis that there is no any significant difference in the population proportions of male with smile and female with smile.
This means we conclude that there is sufficient evidence that there is a significant difference in the population proportions of male with smile and female with smile.
Question 2
Part b
The 95% confidence interval for difference in proportions is given as below:
Confidence interval = (P1 – P) -/+ Z*sqrt[(p1q1/n1)+(p2q2/n2]
The confidence interval by using technology is given as below:
Confidence Interval Estimate
of the Difference Between Two Proportions
Data
Confidence Level
95%
Intermediate Calculations
Z Value
-1.9600
Std. Error of the Diff. between two Proportions
0.0080
Interval Half Width
0.0156
Confidence Interval
Interval Lower Limit
-0.0646
Interval Upper Limit
-0.0333
The above confidence interval support the conclusion in part a as the value ‘0’ is not included in the confidence interval.
Data
Hypothesized Difference
0
Level of Significance
0.05
Group 1
Number of Items of Interest
174
Sample Size
1649
Group 2
Number of Items of Interest
66
Sample Size
1648
Intermediate Calculations
Group 1 Proportion
0.105518496
Group 2 Proportion
0.040048544
Difference in Two Proportions
0.065469952
Average Proportion
0.0728
Z Test Statistic
7.2350
Two-Tail Test
Lower Critical Value
-1.9600
Upper Critical Value
1.9600
p-Value
0.0000
Reject the null hypothesis