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Hypothesis Testing and Estimation

Unit: Quantitative Analysis

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Practice CPA Quantitative Analysis Hypothesis Testing and Estimation questions with detailed answers and explanations.

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December 2025

 1 Questions
Question 3c
​ ​ ​ ​​The immigration department conducted a survey that revealed the monthly number of tourists that visited a certain country in the first six months of year 2025 from eight countries worldwide as recorded in the table below:

Country of originABCDEFGH
Monthly number of tourists252230310360290410380400

Required: 
(i) Test the hypothesis that the number of tourists does not depend on their country of origin at 1% significant level. 

(ii) Interpret the result in (c) (i) above. 


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August 2025

 1 Questions
Question 4b
​ ​ ​ ​​The data below shows the number of students enrolled in six universities for a business course before and after the course was advertised in a local newspaper:

University
Number of students before advertisement
Number of students after advertisement
A
170
180
B
145
151
C
150
143
D
170
177
E
172
178
F
164
167

Required: 

Using the paired t-test, determine whether the advertisement was a success at a 1% level of significance.  


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April 2025

 3 Questions
Question 4a
​​Outline the SIX steps involved in conducting a hypothesis test.


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Question 4b
​ ​ ​​AutoTech Solutions Ltd. is analysing the relationship between total maintenance cost (Y) and two predictor variables: Service hours (X1) and number of replacement parts used (X2). Regression output from Ms Excel is as follows:

Regression statistics:
Parameter
Output
Multiple R
  0.981542 
R square 
A?
Adjusted R square 
  0.955684 
Standard error 
24.137620
Observations
10

ANOVA Table: 
Source
d.f 
SS 
MS
F
Significance
Regression
2
254,600.40
C = ?
D = ?
0.000089
Residual
7
B=?
3,456.21
Total
9
263,000

Regression coefficients: 
Variable
Coefficient
Standard error
t-statics
P value
Lower 95%
Upper 95%
Intercept
480.32
30.1175
E = ?
0.000012
410.28 
550.36
Service hours (​\(X_1\)​) 
F= ?
4.9162
5.847
0.000095
22.32
35.18 
Replacement parts (​\(X_2\)​) 
15.60
3.2453 
48.08
0.000231
10.32
20.88

Required: 

(i) Determine the missing values A, B, C, D, E and F in the regression output.

(ii) Develop a regression equation showing the relationship between total maintenance cost and two predictor variables. 

(iii) Determine the coefficient of determination. Interpret your results. 

(iv) Test the adequacy of the model for prediction (F table value = 4.74). 

(v) Explain whether service hours are adequate as predictor variable (t critical value = 2.365). 


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Question 7a
​​Explain the following terms as used in hypothesis testing: 
 
(i) Type I error. 
 
(ii) Type II error. 
 
(iii) Rejection region.


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December 2024

 1 Questions
Question 7b
​ ​​A company operates under two departments; production and marketing department. Details relating to a sample of employees working in the two departments has been provided as follows:

Accounting Department
Marketing Department
Number of employees
40
50
Average monthly salary 
Sh.120,000
Sh.130,000 
Standard deviation 
Sh.10,000
Sh.12,000

Required: 

Determine whether there is any significant difference between the average salaries of employees working in the two departments. (Use a significance level of 5 percent). 


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August 2024

 2 Questions
Question 6b
​ ​ ​​A manufacturer claims that their packaging machine could fill a packet of sugar with an average weight of 600 grams.

A sample of 36 packets of sugar has an average weight of 500 grams with a standard deviation of 25 grams.

Required:

Test at a 5% significance level if there is sufficient evidence to reject the manufacturers claim


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Question 7c
​ ​​Differentiate between “an interval estimate” and “a point estimate”.


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April 2024

 1 Questions
Question 2b
​ ​ ​​A production manager is convinced that a new set of plant engineers will not affect the production level in the factory. To test this hypothesis, 15 samples of the first day output is taken and average production per day is measured as 14,500 units with a standard deviation of 102. Before the new plant engineers were engaged, the average output per day was 14,650 units.

Required:

Test the manager’s conviction at 5% level of significance


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December 2023

 2 Questions
Question 3c
​ ​​The Revenue Authority Research Department conducted a survey that revealed the number of companies that were reported as being tax compliant in the last six months of the year 2022 as recorded in the table below:

Month
July
August
September
October
November
December
Number of Companies
326
290
306
260
355
365

Required: 

Test the hypothesis that the number of tax compliant companies does not depend on the month of the year at a 1% significance level.


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Question 7a
​ ​Explain the following terms as used in statistics: 

(i) Sample statistic. 

(ii) Population parameter.

(iii) Sampling distribution.


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August 2023

 2 Questions
Question 6a
​​Distinguish between a “two-tailed test” and a “one tailed test” as used in inferential statistics.


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Question 7b
​ ​​Consider the following hypothesis:

​\(H_0: \mu = 400\)​

​\(H_1: \mu \neq 400\)​

For a random sample of 12 observations, the sample mean was 407 and the sample standard deviation was 6.

Required:

Using a significance level of 0.1, advise whether the null hypothesis should be accepted or rejected


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April 2023

 1 Questions
Question 3b
​​The monthly rent paid by residents of Hill View Estate is found to be normally distributed with a mean of Sh.22,150 and a standard deviation of Sh.8,900.

Required:

On the basis of a sample of 200 residents:

(i) Construct a 95% confidence interval for the monthly rent paid.

(ii) Test the hypothesis that the amount paid in monthly rent is Sh.20,000 at a 5% significance level


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December 2022

 1 Questions
Question 4a
​ ​ ​ ​ ​​The Production Manager of Mechtex Ltd., a manufacturer of machines, is investigating a claim by customers about machine A and machine B that it manufactures. 
The claim is that machine A has a longer useful life than machine B. 
A sample of 60 machine As taken from the market reveals that the machine has a mean useful life of 28,000 hours with a standard deviation of 900 hours. A sample of 80 machine Bs has a mean useful life of 30,000 hours with a standard deviation of 1,000 hours. 

Required:

Advise the Production Manager of Mechtex Ltd. if there is a significant difference in the useful lives of the machines. 

Use a significance level of 5%


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August 2022

 3 Questions
Question 6b
​ ​ ​ ​​The weights of 1,000 items are normally distributed with a mean weight of 200 kgs and a standard deviation of
22 kgs.

Required:

(i) Determine the number of items that have weights between 210 kgs and 220 kgs. 

(ii) Determine the number of items that have weights between 180 kgs and 230 kgs. 

(iii) Calculate the weight below which 20% of the items fall.


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Question 3b
​ ​​Differentiate between the following sets of terms as used in hypothesis testing: 

(i) “Null hypothesis” and “alternative hypothesis”. 

(ii) “Parameter” and “statistic”


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Question 1c
​ ​ ​​The number of employees working online during a particular week was recorded as shown below:

Day of the week
Number of employees
Monday
207
Tuesday
185
Wednesday
203
Thursday
180
Friday
225
Total
1,000

Required: 

Test the hypothesis that the number of employees who worked online does not depend on the day of week. Use a significance level of 1%.


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April 2022

 3 Questions
Question 3b
​ ​ ​​The following information relates to the operating life of two brands of mobile phones, brand A and brand B:

Brand A
Brand B
Mean life (days)
1,730
1,684
Standard deviation (in days)
102
108
Sample size
200
200

Required: 

Advise a potential customer on whether there is any significant difference between the quality of the two brands of mobile phones at a 5% level of significance.


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Question 3c
​ ​​A certain new strategy K was claimed to be effective for teams playing a certain game. 

In a marathon of 400 games, half of the teams used strategy K and the other half used strategy P. 

The teams' performance was recorded in the following table

Won
Defeated
Drawn
Adopted strategy K
130
20
60
Adopted strategy P
110
30
50

Required: 
On the basis of the above data and using the chi-square method, advise whether there is a significant difference in the effect of the two strategies, K and P.


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Question 3a
​ ​​A random sample of 350 invoices issued by ARIK Ltd. revealed an average invoice value of Sh.38,780 with a standard deviation of Sh.8,750.

Required:

(i) The interval within which the population mean should fall at 95% level of confidence.

(ii) The sample size of invoices that would result in a 99% confidence level of the interval obtained in (a) (i) above. 


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Pilot December 2021

 2 Questions
Question 3b
​ ​​The manufacturer of the TyroX radical truck tyre claims that the mean mileage the tyre can be driven before the
thread wears out is 60,000 km, assuming the mileage wear follows the normal distribution and a standard deviation
of 5,000 km. In a sample of 48 tyres, the mileage was found to be 59,500 km.

Required:

Test whether this observation is different from the claim by the manufacturer at 5% significance level


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Question 3a
​​Explain the difference between the following terms:

(i) Type 1 and Type II errors.

(ii) One-tail test and two-tail test.

(iii) Normal distribution and t-distribution.


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December 2021

 3 Questions
Question 4b
​ ​ ​ ​​The sales before and after a promotional campaign in ten different regions for a certain commodity were recorded as follows:

Region
Sales before promotional campaign "Sh.million"
Sales after promotional campaign "Sh.million" 
1
53
58
2
28
29
3
31
30
4
48
50
5
50
50
6
42
45
7
63
59
8
40
36
9
25
22
10
30
28

Required: 
Using a 5% level of significance, determine whether the promotional campaign was a success or not. 


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Question 4a
​​In the context of hypothesis testing, distinguish between a "type I error" and a "type II error".


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Question 7a
​ ​​A random sample of 15 employees of a call centre was taken and each employee took a competency test. The mean of the scores achieved by these employees was 56.3% with a standard deviation of 7.1%. The results ofthis test have been found to be normally distributed in the past. 

Required: 

Construct a 95% confidence interval for the mean of the test score of the call centre employees.


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September 2021

 2 Questions
Question 3b(ii)
​ ​ ​ ​​Mali Mali Supermarkets (MMS) operates two branches, one in Kisumu and the other in Mombasa. The management has received complaints that there is a significant difference between the wages paid to employees in the two branches. 

A sample of 40 employees was taken from the Kisumu branch and it had a mean wage of Sh.12,000 and a standard deviation of Sh.1,000. A sample of 50 employees taken from the Mombasa branch had a mean wage of Sh.13,000 and a standard deviation of Sh.1,200. 

Required: 
Advise the management of MMS on whether there is a significant difference between the wages paid in  Kisumu branch and Mombasa branch.

(Use a significance level of 5%).


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Question 3b(i)
​​​​Explain the difference between "point estimate" and "confidence interval". 


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November 2020

 2 Questions
Question 1c
​ ​ ​​An accounting college has two classes, day class and evening class. From a survey conducted by the head of academics in the college, the following results were obtained:

Classes
Day
Evening
Number of students
13
15
Average test mark (%)
45
55
Standard deviation (%)
4
5

Required:
Determine whether there is any significant difference in the average test mark between the two classes at 5% level of significance


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Question 5a
​ ​ ​​With reference to analysis of variances (ANOVA) tests:

(i).   Distinguish between one-way and two-way ANOVA tests.
(ii).  Outline four assumptions of two-way ANOVA tests.
(iii). Explain the difference between ANOVA tests and T-tests.


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November 2019

 1 Questions
Question 2b
​ ​​A manufacturing company is testing a plant for acceptance. For the plant to be accepted, the mean reflectometer reading should be 19.5 and above.

 A random sample of 25 readings is taken and found to have a mean of 19.7 with a standard deviation of 1.5 

Required: 
Test at 95% level of confidence whether the company should accept the new plant.


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May 2019

 1 Questions
Question 3b
​  ​(i)  In relation to hypothesis testing and estimation, distinguish between "null hypothesis" and "alternative hypothesis". 
(ii)  Beta Limited deals in the manufacture of a detergent named "safi". A recent survey undertaken to determine the percentage of residents who use "safi" revealed that out of 500 residents selected at random, only 10% used "safi". In order to increase the usage of "safi" amongst the residents, the company undertook an advertising campaign that cost Sh.2.5 million. A survey undertaken after the campaign revealed that out of 600 residents selected at random, 15% used "safi". 

Required: 
Determine whether the advertising campaign increased the usage of "safi" amongst the residents. (Use a significance level of 5%).


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November 2018

 2 Questions
Question 1c
​​A company operates under two departments, P and Q. Details relating to a sample of employees working in the two departments has been provided as follows:

Department P
Department Q
Number of employees
29
24
Average monthly salary
Sh.260,000
Sh.310,000
Standard deviation
Sh.25,000
Sh.30,000

Required: 
Determine whether there is any significant difference between the average monthly salaries of employees working in the two departments. (Use a significance level of 5 per cent).


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Question 3a
​ ​​ Explain the following terms as used in hypothesis testing: 

(i) Level of significance. 
(ii) Region of rejection.


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May 2018

 3 Questions
Question 3b
​​Faida Limited deals in the manufacture and sale of a product named "Big". The company sells the product in two of its distribution outlets, A and B. 

The data below relate to a random survey of monthly mean sales of the product in the two outlets:

Outlet
Monthly mean sales
(Sh."000")
Standard deviation
(Sh."000")
Sample size
A
795
50
200
B
810
70
175

Required:
Test at a 5 per cent level of significance, whether there is a significant difference in the monthly mean sales of the two outlets. 


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Question 3a
​​Explain the difference between the following sets of terms as used in hypothesis testing and estimation:
(i)  "Type 1 error" and "type II error". 
(ii) "One tailed test" and "two tailed test".



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Question 2c
​ ​​Mwanaisha Ali sells tree seedlings at Mavuno market. A random sample of 9 of the seedlings had the following height in centimetres:

64  62  65  63  68  69  65  63  65.

Required: 
A 95 per cent confidence interval of the population mean height of the seedlings.


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December 2017

 1 Questions
Question 2b
​ ​​A company produces two types of electric components whose information has been provided to you as follows:

Component X
Component Y
Mean life in hours
1,600
1,528
Standard deviation in hours
132
149
Sample size
120
110

Required: 
Determine whether the quality of the two types of electric components differ significantly. (Use a significance level of 95%). 


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November 2016

 1 Questions
Question 1c
​​The data below show the number of students enrolled in six colleges for a certain course, before and after the course was advertised in a certain publication:
College
Number of students
 before advertisement
Number of students
after advertisement
1
165
170
2
140
141
3
143
142
4
160
167
5
162
168
6
154
157

Required:
Using the paired t-test, determine whether the advertisement was a success at a 5 per cent level of significance.


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May 2016

 1 Questions
Question 5a
​​Outline three differences between the normal distribution and the t-distribution.


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