Question

Never forget that even small effects can be statistically significant if the samples are large. To illustrate this fact, consider a sample of 112 small businesses. During a three-year period, 10 of the 73 headed by men and 7 of the 39 headed by women failed.

(a) Find the proportions of failures for businesses headed by women and businesses headed by men. These sample proportions are quite close to each other. Give the P-value for the test of the hypothesis that the same proportion of women's and men's businesses fail. (Use the two-sided alternative). What can we conclude (Use α=0.05α=0.05)?
The P-value was so we conclude that
Choose a conclusion.The test showed strong evidence of a significant difference.The test showed no significant difference.

(b) Now suppose that the same sample proportion came from a sample 30 times as large. That is, 210 out of 1170 businesses headed by women and 300 out of 2190 businesses headed by men fail. Verify that the proportions of failures are exactly the same as in part (a). Repeat the test for the new data. What can we conclude?
The P-value was so we conclude that
Choose a conclusion.The test showed strong evidence of a significant difference.The test showed no significant difference.

(c) It is wise to use a confidence interval to estimate the size of an effect rather than just giving a P-value. Give 95% confidence intervals for the difference between proportions of men's and women's businesses (men minus women) that fail for the settings of both (a) and (b). (Be sure to check that the conditions are met. If the conditions aren't met for one of the intervals, use the same type of interval for both)
Interval for smaller samples: to
Interval for larger samples: to

What is the effect of larger samples on the confidence interval?
Choose an effect.The confidence interval is unchanged.The confidence interval's margin of error is reduced.The confidence interval's margin of error is increased.

please show your work and what function you used, if any on the calculator thank you !

Answer 1

using excel>addin>phstat>two sample test

we have

Z Test for Differences in Two Proportions
Data
Hypothesized Difference	0
Level of Significance	0.05
women
Number of Items of Interest	7
Sample Size	39
men
Number of Items of Interest	10
Sample Size	73
Intermediate Calculations
women Proportion	0.179487179
men Proportion	0.136986301
Difference in Two Proportions	0.042500878
Average Proportion	0.1518
Z Test Statistic	0.5972
Two-Tail Test
Lower Critical Value	-1.9600
Upper Critical Value	1.9600
p-Value	0.5504
Do not reject the null hypothesis

(a) the proportions of failures for businesses headed by women and businesses headed by men are 0.18 and 0.137 respectively These sample proportions are quite close to each other. Give the P-value for the test of the hypothesis that the same proportion of women's and men's businesses fail. (Use the two-sided alternative). What can we conclude (Use α=0.05α=0.05)?
The P-value was 0.5504 which is greater than 0.05 so we conclude that
Choose a conclusion.The test showed no significant difference.

(b)using excel>addin>phstat>two sample test

Z Test for Differences in Two Proportions
Data
Hypothesized Difference	0
Level of Significance	0.05
women
Number of Items of Interest	210
Sample Size	1170
men
Number of Items of Interest	300
Sample Size	2190
Intermediate Calculations
Group 1 Proportion	0.179487179
Group 2 Proportion	0.136986301
Difference in Two Proportions	0.042500878
Average Proportion	0.1518
Z Test Statistic	3.2710
Two-Tail Test
Lower Critical Value	-1.9600
Upper Critical Value	1.9600
p-Value	0.0011
Reject the null hypothesis

yes the proportions of failures are exactly the same as in part (a).
The P-value was 0.0011 which is less than 0.05 so we conclude that The test showed strong evidence of a significant difference.

(c)For small sample

Z Test for Differences in Two Proportions
Data		Confidence Interval Estimate
Hypothesized Difference	0		of the Difference Between Two Proportions
Level of Significance	0.05
Group 1			Data
Number of Items of Interest	10		Confidence Level	95%
Sample Size	73
Group 2			Intermediate Calculations
Number of Items of Interest	7		Z Value	-1.9600
Sample Size	39		Std. Error of the Diff. between two Proportions	0.0735
			Interval Half Width	0.1440
Intermediate Calculations
Group 1 Proportion	0.136986301		Confidence Interval
Group 2 Proportion	0.179487179		Interval Lower Limit	-0.1865
Difference in Two Proportions	-0.04250088		Interval Upper Limit	0.1015
Average Proportion	0.1518
Z Test Statistic	-0.5972
Two-Tail Test
Lower Critical Value	-1.9600
Upper Critical Value	1.9600
p-Value	0.5504
Do not reject the null hypothesis

For large sample

Z Test for Differences in Two Proportions
Data		Confidence Interval Estimate
Hypothesized Difference	0		of the Difference Between Two Proportions
Level of Significance	0.05
Group 1			Data
Number of Items of Interest	300		Confidence Level	95%
Sample Size	2190
Group 2			Intermediate Calculations
Number of Items of Interest	210		Z Value	-1.9600
Sample Size	1170		Std. Error of the Diff. between two Proportions	0.0134
			Interval Half Width	0.0263
Intermediate Calculations
Group 1 Proportion	0.136986301		Confidence Interval
Group 2 Proportion	0.179487179		Interval Lower Limit	-0.0688
Difference in Two Proportions	-0.04250088		Interval Upper Limit	-0.0162
Average Proportion	0.1518
Z Test Statistic	-3.2710
Two-Tail Test
Lower Critical Value	-1.9600
Upper Critical Value	1.9600
p-Value	0.0011
Reject the null hypothesis

Interval for smaller samples: -0.1865 to 0.1015
Interval for larger samples:-0.0688 to -0.0162

What is the effect of larger samples on the confidence interval?
Choose an effect.The confidence interval's margin of error is reduced.