Question

Consider the following data drawn independently from normally distributed populations: (You may find it useful to reference the appropriate table: z table or t table)

x−1x−1 = 25.7	x⎯⎯2x¯2 = 30.6
σ12 = 98.2	σ22 = 87.4
n1 = 20	n2 = 25

a. Construct the 95% confidence interval for the difference between the population means. (Negative values should be indicated by a minus sign. Round all intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.)

Confidence internval is _____ to ____.

Consider the following competing hypotheses and accompanying sample data drawn independently from normally distributed populations. (You may find it useful to reference the appropriate table: z table or t table)

H₀: μ₁ − μ₂ = 0
H_A: μ₁ − μ₂ ≠ 0

x−1x−1 = 51	x−2x−2 = 60
σ1 = 13.00	σ2 = 1.64
n1 = 25	n2 = 25

a-1. Calculate the value of the test statistic. (Negative values should be indicated by a minus sign. Round all intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)
  

Test statistic:

Researchers at The Wharton School of Business have found that men and women shop for different reasons. While women enjoy the shopping experience, men are on a mission to get the job done. Men do not shop as frequently, but when they do, they make big purchases like expensive electronics. The accompanying table shows the amount spent (in $) over the weekend by 40 men and 60 women at a local mall. The Excel file is also provided. (You may find it useful to reference the appropriate table: z table or t table)

Spending by Men	Spending by Women	Spending by Men	Spending by Women
85	90	87	38
102	79	92	66
139	71	92	100
90	119	72	57
89	90	97	59
52	180	83	89
49	88	118	95
140	56	108	37
90	110	104	86
64	82	110	62
96	64		66
132	129		129
117	28		119
88	13		76
92	140		75
105	62		101
95	32		85
119	220		68
118	72		67
124	90		36
131	80		90
113	56		99
124	82		64
71	56		54
115	88		86
95	104		79
102	54		82
94	108		65
111	86		110
85	88		69

Click here for the Excel Data File

Let µ₁ represent the population mean amount spent by men and µ₂ represent the population mean amount spent by women.

a. Specify the competing hypotheses that determine if the mean amount spent by men is more than that by women.

• H₀: μ₁ − μ₂ = 0; H_A: μ₁ − μ₂ ≠ 0

• H₀: μ₁ − μ₂ ≥ 0; H_A: μ₁ − μ₂ < 0

• H₀: μ₁ − μ₂ ≤ 0; H_A: μ₁ − μ₂ > 0

b. Calculate the value of the test statistic. Assume that the population variances are unknown but equal. (Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)

test statistic:

Answer 1

using excel>addin>phstat>two sample test

we have

Pooled-Variance t Test for the Difference Between Two Means
(assumes equal population variances)
Data		Confidence Interval Estimate
Hypothesized Difference	0		for the Difference Between Two Means
Level of Significance	0.05
Population 1 Sample			Data
Sample Size	20		Confidence Level	95%
Sample Mean	25.7
Sample Standard Deviation	9.91		Intermediate Calculations
Population 2 Sample			Degrees of Freedom	43
Sample Size	25		t Value	2.0167
Sample Mean	30.6		Interval Half Width	5.8086
Sample Standard Deviation	9.349
			Confidence Interval
Intermediate Calculations		Interval Lower Limit	-10.7086
Population 1 Sample Degrees of Freedom	19		Interval Upper Limit	0.9086
Population 2 Sample Degrees of Freedom	24
Total Degrees of Freedom	43
Pooled Variance	92.1778
Standard Error	2.8803
Difference in Sample Means	-4.9000
t Test Statistic	-1.7012
Two-Tail Test
Lower Critical Value	-2.0167
Upper Critical Value	2.0167
p-Value	0.0961
Do not reject the null hypothesis

a ) The confidence interval is -10.71 to 0.91

Ans 2 )

H0: μ1 − μ2 = 0
HA: μ1 − μ2 ≠ 0

using excel>addin>phstat>two sample test

we have

Pooled-Variance t Test for the Difference Between Two Means
(assumes equal population variances)
Data
Hypothesized Difference	0
Level of Significance	0.05
Population 1 Sample
Sample Size	25
Sample Mean	51
Sample Standard Deviation	13
Population 2 Sample
Sample Size	25
Sample Mean	60
Sample Standard Deviation	1.64
Intermediate Calculations
Population 1 Sample Degrees of Freedom	24
Population 2 Sample Degrees of Freedom	24
Total Degrees of Freedom	48
Pooled Variance	85.8448
Standard Error	2.6206
Difference in Sample Means	-9.0000
t Test Statistic	-3.4343
Two-Tail Test
Lower Critical Value	-2.0106
Upper Critical Value	2.0106
p-Value	0.0012
Reject the null hypothesis

a-1. the value of the test statistic t = -3.43

Ans3 ) Researchers at The Wharton School of Business have found that men and women shop for different reasons. While women enjoy the shopping experience, men are on a mission to get the job done. Men do not shop as frequently, but when they do, they make big purchases like expensive electronics. The accompanying table shows the amount spent (in $) over the weekend by 40 men and 60 women at a local mall. The Excel file is also provided. (You may find it useful to reference the appropriate table: z table or t table)

Click here for the Excel Data File

Let µ1 represent the population mean amount spent by men and µ2 represent the population mean amount spent by women.

a. Specify the competing hypotheses that determine if the mean amount spent by men is more than that by women.

• H0: μ1 − μ2 ≤ 0; HA: μ1 − μ2 > 0

usig excel we have

Pooled-Variance t Test for the Difference Between Two Means
(assumes equal population variances)
Data
Hypothesized Difference	0
Level of Significance	0.05
Population 1 Sample
Sample Size	40
Sample Mean	99.75
Sample Standard Deviation	21.35386
Population 2 Sample
Sample Size	60
Sample Mean	82.1
Sample Standard Deviation	33.86899
Intermediate Calculations
Population 1 Sample Degrees of Freedom	39
Population 2 Sample Degrees of Freedom	59
Total Degrees of Freedom	98
Pooled Variance	872.0704
Standard Error	6.0280
Difference in Sample Means	17.6500
t Test Statistic	2.9280
Upper-Tail Test
Upper Critical Value	1.6606
p-Value	0.0021
Reject the null hypothesis

b. the value of the test statistic = 2.93