Hypothesis Testing and Confidence Intervals

Hypothesis Testing and Confidence Intervals

Mean Sales Per Week

Hypothesis

We test the hypotheses;

H₀: µ ≤ 42.5

H_a: µ > 42.5

The level of significance is alpha = 0.05

Table 1: One-Sample t-test for mean sales per week

t-Test: one-Sample
	Sales (Y)
Mean	41.89
Variance	70.32111111
Observations	100
Hypothesized Mean	42.5
df	99
t Stat	-0.727422907
P(T<=t) one-tail	0.234341858
t Critical one-tail	1.660391156
P(T<=t) two-tail	0.468683716
t Critical two-tail	1.984216952

The hypothesis test is one-tailed, more specifically, right-tailed. Therefore, we reject the null hypothesis if the test statistic value is greater than the critical value. The table indicates that the resulting p-value is 0.2343, which is greater than the level of significance, 0.05. Thus, we fail to reject the null hypothesis and conclude that mean sales per week per person does not exceed 42.5 at a 5% level of significance.

Confidence Intervals

The 99% confidence interval for the mean sales per week is given by:

Where:

Therefore, we are 99% confident that the true mean lies between 39.6834 and 44.0966.

Proportion Receiving Online Training

Hypothesis Test

We test the hypotheses;

H₀: p ≥ 0.55

H_a: p < 0.55

Alpha = 0.05

Standard Error =

Z =  

Table 2: Proportion break-down

Observed Proportion	0.53
Hypothesized Value	0.55
Standard error	0.049749372
Z statistic	-0.402015126
Z critical value (one-tailed)	-1.645
P-value	0.34458
Alpha	0.05
Decision	Fail to reject the null hypothesis

The test is left-tailed, and thus, we reject the null hypothesis when the test statistic is less than the critical value. Since the resulting Z statistic value of -0.40 is greater than the critical value, we fail to reject the null hypothesis. The implication is that the proportion of people receiving online training is not less than 55%, at a 95% confidence level.

Confidence Level

The 99% confidence level for the proportion of individuals receiving online training is given by;

=

             

= [0.40, 0.66]

We are 99% confident that the proportion of people receiving online training lies between 40% and 66%.

Mean Calls Made Among Those with No Training

Hypothesis Test

We test the hypotheses;

H₀: µ ≤ 145

H_a: µ > 145

Alpha = 0.05

Table two: One-Sample t-test for

t-Test: One-Sample
	Calls (X1)
Mean	140.8888889
Variance	94.33986928
Observations	18
Hypothesized Mean	145
Df	17
t Stat	-1.795758092
P(T<=t) one-tail	0.04516418
t Critical one-tail	1.739606726
P(T<=t) two-tail	0.09032836
t Critical two-tail	2.109815578

The test is right-tailed, and thus, we reject the null hypothesis if the test statistic exceeds the critical value. Since the absolute value of the t statistic, 1.796 is greater than 1.740, we reject the null hypothesis. The mean calls made among those with no training exceeds 145.

Confidence interval

The 99% confidence interval is given by;

where

= [138.074, 143.704]

We are 99% confident that the mean calls made per week per person lie between 138.074 and 143.704.

Mean Time Per Call

Hypothesis Test

We test the hypotheses;

H₀: µ = 14.7

H_a: µ ≠ 14.7

Alpha = 0.05

Table Four: One-sample t-test for mean time per call

t-test: One-Sample
	Time (X2)
Mean	15.226
Variance	5.423155556
Observations	100
Hypothesized Mean	14.7
df	99
t Stat	2.25870604
P(T<=t) one-tail	0.013047659
t Critical one-tail	1.660391156
P(T<=t) two-tail	0.026095317
t Critical two-tail	1.984216952

Since the resulting t-statistic value of 2.2587 is greater than 1.9842, we reject the null hypothesis and conclude that the mean time per call is significantly different from 14.7.

Confidence Interval

Where:

 Hence, with 99% confidence, the true mean time per call lies between 14.613 and 15.839.

Conclusion

The mean sales per week per person was found not to exceed 42.5 at a 95% confidence level. The corresponding 99% confidence level was also found to lie between 39.6834 and 44.0699. However, the proportion of the individuals receiving online is less than 55%. On the other hand, mean calls made among those with no training is less than 145, and the mean time per call is significantly different from 14.7.

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The attachment is the data set on the project data sales tab. Then, each title is broken down to show the mean, standard deviation, etc. Instructions below.
Complete the following four hypotheses, using α = 0.05 for each. The week 5 spreadsheet can be used in these analyses.

1. Mean sales per week exceed 42.5 per salesperson
2. The proportion receiving online training is less than 55%
3 Mean calls made among those with no training is at least 145
4. Mean time per call is 14.7 minutes
Using the same data set from part A, perform the hypothesis test for each speculation in order to see if there is evidence to support the manager’s belief. Use the Eight Steps of a Test of Hypothesis from Section 9.1 of your textbook as a guide. You can use either the p-value or the critical values to draw conclusions. Be sure to explain your conclusion and interpret the claim in simple terms.
Compute 99% confidence intervals for the variables used in each hypothesis test and interpret these intervals.
Write a report about the results, distilling down the results in a way that would be understandable to someone who does not know statistics. Clear explanations and interpretations are critical.
All DeVry University policies are in effect, including the plagiarism policy.
Project Part B report is due by the end of Week 6.
Project Part B is worth 100 total points. See the grading rubric below.
Format for report:
Summary Report (about one paragraph on each of the four speculations)
Appendix with the calculations of the Eight Elements of a Test of Hypothesis, the p-values, and the confidence intervals. Include the Excel formulas or spreadsheet screenshots used in the calculations.