Reference no: EM132314874

Statistical Inference Assignment -

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

Q1. State whether the following statements are True or False. Give reason in support of your answer:

(a) If the probability density function of a random variable X follows t-distribution is f(x) = 1/(π(1+x²)), x ≥ 0 then degrees of freedom of the distribution will be 2.

(b) The cars that enter in a Metro parking are classified either Indian-made or Foreign-made. To check that the car ender in the parking is in random order, we use Mann Whitney U test.

(c) A random variable has the pdf f(x) = 1/θ, 0 ≤ x ≤ θ

If the critical region of testing the null and alternative hypotheses H₀: θ = 2 and H₀: θ ≠ 2 is X > 1 then type-I error will be 0.5.

(d) If sample mean (X^-) is consistent estimator of the parameter θ then log(X^-) also consistent for log(θ).

(e) In a random sample of 525 families owning television set in the region of New Delhi, it is found that 370 subscribe to Star Plus. A 99% confidence interval for the actual proportion of such families in New Delhi which subscribe to Star Plus will be (0.66, 0.74).

Q2. A baby-sister has 6 children under her supervision. The age of each child is as follows:

Child	Age (in years)
Sonu	10
Rishi	8
Lavnik	6
Chiya	4
Aman	2
Avishi	6

i) Find the mean and SD of this finite population.

ii) List all possible sample of size 3 from this population without replacement.

iii) Construct the sampling distribution of mean.

iv) Compute the mean and standard error of the mean of the sampling distribution obtained in (iii).

Q3. (a) A Pizza company would like to determine the average delivery time it can promise its customers. How large should the sample size be if it wants to be 95% confident that the sample estimate would not differ from the actual average delivery time by more than 1.5 minutes? The previous studies have shown the SD to be 7 minutes.

(b) A sample of 400 shops was selected in a large metropolitan area to determine various information concerning to the consumer behaviour. One question, among the questions, asked, was "Do you enjoy shopping for clothing?" Out of 200 males, 170 answered yes. Out of 250 females, 224 answered yes. Find 95% confidence interval for the difference of the proportions for enjoys shopping for clothing.

Q4. The following data relate to the number of items produced in a shift by two workers A and B for some days:

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

Assuming that the parent populations are normal, can it be inferred that B is more stable or consistent) worker compared to A?

Q5. An economist wants to undertake a survey to establish if there is any relationship between the age of the person and his/her attitude towards the economy of the nation during current administration as compared to the economy during the previous administration. A random sample of 500 persons across the country was selected and they were put into categories of their age group and their respective opinions regarding the economy. These categories are shown in the following table:

Age Group	Opinion on Economy
	Improve	Remained Same	Worsened
Below 30 years	40	64	96
30 to 50 years	84	52	24
Above 5o years	30	40	70

Test whether the age group and attitude towards economy are dependent at 5 % level of significance?

Q6. Complete the following table, one is done for you:

S.No.	Test For	Name of the Test	Test Statistic	Assumptions for Applying the Test	Test Type
1	Population mean when population variance is known and population is normal	Z-test	Z = (X- - μ)/σ√n	1. Sample observations should be independent. 2. The measurement scale should be at least interval scale.	Parametric Test
2	Population mean when population variance is unknown and population is normal
3	Two population means/medians when the form of populations is normal, samples are independent, population SDs are known and sample size is small
4	Two population means/medians when the form of populations is normal, samples are independent, population SDs are unknown and sample size is small
5	Two population means/medians when the form of populations is not known, samples are independent and sample size is small
6	Independence of Two Attributes

Q7. If the magnitude of the earthquakes recorded in a region of a country follows a distribution with parameter θ

f(x) = (1/θ²) xe^-x/θ, x ≥ 0,  θ ≥ 0

then find

i) Maximum likelihood estimator of the parameter θ,

ii) Maximum likelihood estimate of the parameter on the basis of the following data:1

Magnitude of the Earthquakes (on the Richter scale)

6.7

7.7

5.6

7.3

6.7

6.6

7.8

6.7

6.2

5.2

iii) Show that maximum likelihood estimator is unbiased and sufficient for parameter θ.

Q8. A medical insurance company is encouraging its subscribers to buy generic drugs because they are cheaper rather than brand name drugs. In order to test the company's claim, random samples of prices of 8 different drugs are compared. The following table represents the prices of both types of drugs bought from eight different pharmacies at random:

Pharmacy	Generic Drug	Brand name Drug
1	20	21
2	8	11
3	15	15
4	32	40
5	15	22
6	12	15
7	18	17
8	20	25

i) Formulate the hypotheses to test the claim

ii) What assumptions are necessary to apply the parametric test to test the claim.

iii) Apply the parametric test for testing the hypotheses formulated in (i) and under the assumptions (ii) using α = 0.05.

iv) If assumptions in (ii) are not fulfilled, then which test is applied in this situation and why?

v) Apply the test for (iv) and compare the result with (iii).