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MST 4 2025 - English

TUTOR MARKED ASSIGNMENT

MST-004: Statistical Inference

Course Code: MST-004

Assignment Code: MST-004/TMA/2025

Maximum Marks: 100

Note: All questions are compulsory. Answer in your own words.

1. State whether the following statements are True or False. Give reason in support of your answer:

(a) If the probability of non rejection of Ho when H, is true is 0.4 then power of the test will be 0.6.

(b) If T₁ and T₂ are two estimators of the parameter & such that Var(T₁)-1/n and Var(T₂)n then T, is more efficient than T₂.

(c) A 95% confidence interval is smaller than 99% confidence interval.

(d) If the level of significance is the same, the area of the rejection region in a two-tailed test is less than that in a one-tailed test.

(e) Non parametric tests are more powerful than the parametric tests.

2. If a finite population has four elements: 6, 1, 3, 2.

(a) How many different samples of size n=2 can be selected from this population if you sample without replacement?

(b) List all possible samples of size n=2.

(c) Compute the sample mean for each of the samples given in part b.

(d) Find the sampling distribution off and draw the histogram.

(e) Compute standard error.

(f) If all four population values are equally likely, calculate the value of the population mean ? Do any of the samples listed in part (b) produce a value off exactly equal to

3. A study was conducted to compare the mean numbers of police emergency calls per 8-hour shift in two districts of a large city. Samples of 100 8-hour shifts were randomly selected from the police records for each of the two regions and the number of emergency calls was recorded for each shift. The sample statistics are listed here:

	Region
	1	2
Sample size	100	100
Sample mean	2.4	3.1
Sample variance	1.44	2.64

Find a 90% confidence interval for the difference in the mean numbers of police emergency calls per shift between the two districts of the city. Interpret the interval.

4. A bond proposal for school construction will be submitted to the voters at the next municipal election. A major portion of the money derived form this bond issue will be used to build schools in a rapidly developing selection of the city, and the remainder will be used to renovate and update school buildings in the rest of the city. To assess the viability of the bond proposal, a random sample of-50 residents in the developing section and n100 residents from the other parts of the city were asked whether they plan to vote for the proposal. The results are tabulated below

Sample Values for Opinion on Bond Proposal

	Developing Section	Rest of the City
Sample size	50	100
Number favoring proposa	38	65

(a) Estimate the difference in the true proportions favoring the bond proposal with a 99% confidence interval.

(b) If both samples were pooled into one sample of size n-150, with 103 in favor of the proposal, provide a point estimate of the proportion of city residents who will vote for the bond proposal.

5. The following data relate to the number of items produced per shift by two workers for a number of days:

Worker A	19	22	24	27	24	18	20	19	25
Worker B	26	37	40	35	30	40	26	30	35	45

6. If magnitude of earthquakes recorded in a region of a country follows a distribution with parameter a whose pdf is given below:

then show that the estimators of the parameter u using maximum likelihood and method of moments are same.

7. A company plans to promote a new product by using one of three advertising campaigns. To investigate the extent of product recognition from these three campaigns, 15 market areas were selected and five were randomly assigned to each advertising plan. At the end of the ad campaigns, random samples of 400 adults were selected in each area and the proportions who were familiar with new product were recorded. The responses were not approximately normal. Is there a significant difference among the three population distributions from which these samples came? Use an appropriate nonparametric method to answer this question at 5% level of significance. then show that the estimators of the parameter u using maximum likelihood and method of moments are same.

Campaign
1	2	3
0.33	0.28	0.21
0.29	0.41	0.30
0.21	0.34	0.26
0.32	0.39	0.33
0.25	0.27	0.31

8.A psychology class performed an experiment to determine whether a recall score in which instructions to form images of 25 words were given differs from an initial recall score for which no imagery instructions were given. Twenty students participated in the experiment with the results listed in the table:

Student	With Imagery	Without Imagery	Student	With Imagery	Without Imagery
1	20	5	11	17	8
2	24	9	12	20	16
3	20	5	13	20	10
4	18	9	14	16	12
5	22	6	15	24	7
6	19	11	16	22	9
7	20	8	17	25	21
8	19	11	18	21	14
9	17	7	19	19	12
10	21	9	20	23	12

(a) What two testing procedures can be used to test for differences in the distribution of recall scores with and without imagery? What assumptions are required for the parametric procedure? Do these data satisfy these assumptions?

(b) Use both the parametric and non-parametric tests for differences in the distributions of recall scores under these two conditions.

(c) Compare the results of the tests in part b. Are the conclusions for same? If not, why not?

MST 004 (January 2026 - July 2026) - ENGLISH

TUTOR MARKED ASSIGNMENT

MST-004: Statistical Inference

Course Code: MST-004

Assignment Code: MST-004/TMA/2026

Maximum Marks: 100

Note: All questions are compulsory. Answer in your own words.

1. State whether the following statements are True or False. Give reason in support of your answer:

(a) The Wilcoxon rank-sum test is a non-parametric alternative to the two-sample t-test when normality is not assumed.

(b) The power of a statistical test is the probability of accepting the null hypothesis when it is true.

(c) If the sample size tends to infinity, the variance of a consistent estimator must tend to zero.

(d) In the method of moments, population moments are equated to corresponding sample moments.

(e) The chi-square distribution is symmetric for all degrees of freedom.

2. A school caretaker looks after 6 children. The ages (in years) of the children are given below:

Child Name	Age (in years)
Meena	5
Aarav	4
Nisha	6
Kabir	4
Pooja	5

(i) What is the nature (type) of the population of the ages of children?

(ii) Construct the sampling distribution of the sample mean for all possible samples of size .

(iii) Comment on whether the sampling distribution is approximately normal.

(iv) Find the mean and standard error of the sampling distribution. 

3. Let be a random sample from a population with mean and variance . Consider the estimator
 

(i) Find the bias of the estimator .

(ii) Find the variance of .

(iii) Check whether it is more efficient than sample mean. 

4. A survey was conducted to study the preference for online learning among students from urban and rural areas. Out of 300 urban students, 210 preferred online learning, while out of 250 rural students, 150 preferred online learning. Construct a 95% confidence interval for the difference between the population proportions of students preferring online learning in urban and rural areas. 

5. A researcher wants to compare the average test scores of students taught using Method A and Method B. A random sample of students was selected from each group, and their scores are given below:

Student	Method A	Method B
1	68	72
2	74	75
3	71	78
4	69	70
5	73	76
6	70	74
7	72	77
8	75	79

Assuming that the populations are normally distributed with equal variances. For testing whether there is a significant difference between the mean scores of the two teaching methods at the 5% level of significance

(i) State the null and alternative hypotheses.

(ii) Name the appropriate and justify your choice.

(iii) Compute the test statistic and critical value.

(iv) State your conclusion. 

6. Let  be a random sample from the Exponential distribution with pdf

(i) Obtain the maximum likelihood estimator (MLE) of .

(ii) Hence, find the maximum likelihood estimate of  for the observed sample: 

2.1, 1.6, 3.4, 0.9, 2.5

7. Describe the following:

(i) t-distribution

(ii) General procedure of Testing of hypothesis 

 

 

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