When the alternative hypothesis is $H_1 : \mu \ne \mu_0$, the critical region will be determined by :
a.
right tail
b.
both right and left tails
Correct Answer
c.
left tail
d.
neither right nor left tail
Step-by-Step AI Explanation
Verified Method
When $H_1: \mu \ne \mu_0$, it is a two-tailed test, so the critical region lies in both the right and left tails of the sampling distribution. Hence, option (b) is correct. Options (a) and (c) are for one-tailed tests, and (d) is incorrect because there must be a rejection region in a two-sided test.
Q2mcq1 mark
What is level of significance ?
a.
Upper limit of $P$ (Type I Error)
Correct Answer
b.
$P$ (Type I Error)
c.
Upper limit of $P$ (Type II Error)
d.
$P$ (Type II Error)
Step-by-Step AI Explanation
Verified Method
The level of significance is the upper limit of the probability of committing Type I error, that is, rejecting a true null hypothesis. It is usually denoted by $\alpha$. Therefore, option (a) is correct. Options (b), (c), and (d) are incorrect because Type II error is denoted by $\beta$, not the level of significance.
Q3mcq1 mark
If $n$ is the degree of freedom of chi-square distribution then its variance is :
a.
$2n$
Correct Answer
b.
$n$
c.
$n11$
d.
$n-1$
Step-by-Step AI Explanation
Verified Method
For a chi-square distribution with $n$ degrees of freedom, the variance is $2n$. So the correct answer is (a). The mean is $n$, not the variance, and the other options are incorrect.
Q4mcq1 mark
In two-way classification the total variation TSS =
a.
$SST1SSB-SSE$
b.
$SST1SSB1SSE$
c.
$SST1SSB$
Correct Answer
d.
$SST-SSB1SSE$
Step-by-Step AI Explanation
Verified Method
In two-way classification, the total sum of squares is partitioned into row variation and column variation. Hence, $TSS = SST + SSB$. Therefore, option (c) is correct. The other options include the error sum of squares incorrectly or do not represent the correct partition.
Q5mcq1 mark
F-statistic is the ratio of two Independent sample __________.
a.
Standard deviation
b.
Mean
c.
Mode
d.
Variances
Correct Answer
Step-by-Step AI Explanation
Verified Method
The F-statistic is defined as the ratio of two independent sample variances. Therefore, option (d) is correct. It is not the ratio of means, standard deviations, or modes.
Q6mcq1 mark
If $\Sigma D^2 = 0$, rank correlation is :
a.
0.5
b.
0
c.
−1
d.
1
Correct Answer
Step-by-Step AI Explanation
Verified Method
For Spearman's rank correlation coefficient, if $\sum D^2 = 0$, then there is no difference between the paired ranks, so the correlation is perfect positive, i.e. $1$. Hence, option (d) is correct.
Q7mcq1 mark
If $r=0$ then $\operatorname{cov}(x,y)$ is :
a.
−1
b.
0
Correct Answer
c.
α
d.
11
Step-by-Step AI Explanation
Verified Method
Since correlation coefficient $r=0$ indicates no linear relationship, covariance is also $0$ because $r = \dfrac{\operatorname{cov}(x,y)}{\sigma_x\sigma_y}$. Therefore, option (b) is correct.
Q8mcq1 mark
__________ is the Father of mental tests.
a.
Francis Galton
Correct Answer
b.
R.A. Fisher
c.
A.L. Bowley
d.
Croxton and Cowden
Step-by-Step AI Explanation
Verified Method
Francis Galton is known as the Father of mental tests. Hence, option (a) is correct. R.A. Fisher is associated with modern statistical methods, while A.L. Bowley and Croxton & Cowden are not called the father of mental tests.
Q9mcq1 mark
The regression lines intersect at :
a.
$(0,0)$
b.
$(X,Y)$
Correct Answer
c.
$(1,1)$
d.
$(X,Y)$
Step-by-Step AI Explanation
Verified Method
The two regression lines always intersect at the point of means $({\bar X},{\bar Y})$. Hence the correct choice is $(X,Y)$, i.e. option **(b)**. Options (a) and (c) are not generally true, and option (d) repeats the same point.
Q10mcq1 mark
The condition for the circular test to hold good with usual notation is :
a.
$P_{01}÷P_{10}=1$
b.
$P_{01}×P_{10}=1$
c.
$P_{01}1P_{12}1P_{20}=1$
d.
$P_{01}×P_{12}×P_{20}=1$
Correct Answer
Step-by-Step AI Explanation
Verified Method
For the circular test to hold good, the product of the price relatives in a complete cycle must be 1. With usual notation, the condition is
$$P_{01}\times P_{12}\times P_{20}=1.$$
So the correct answer is **(d)**. The other options do not represent the circular test condition.
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a.
Rs. 1,18,000 p.a.
b.
Rs. 1,08,000 p.a.
c.
Rs. 1,80,000 p.a.
d.
Rs. 1,02,000 p.a.
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The mean breaking strength of cables supplied by a manufacturer is 1900 n/m² with a Standard Deviation of 120 n/m². The manufacturer introduced a new technique in the manufacturing process and claimed that the breaking strength of the cables has increased. In order to test the claim, a sample of 60 cables is tested. It is found that the mean breaking strength of the sampled cables is 1960 n/m². Can we support the claim at 1% level of significance?
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The number of pages typed by 5 DTP-Operators for 1 hour in the morning sessions are 10, 12, 13, 8, 9 and the number of pages typed by them for 1 hour in the afternoon are 11, 15, 12, 10, 8. Is there any significant difference in the mean number of pages typed?
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Three different techniques namely medication, exercises and special diet are randomly assigned to individuals diagnosed with high blood pressure to lower the blood pressure. After four weeks the reduction in each person's blood pressure is recorded. Test at 5% level, whether there is significant difference in mean reduction of blood pressure among the three techniques.
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The following table gives quarterly expenditure over a number of years. Obtain seasonal indices for the data.
Year
Season
I 78 84 92 100
II 62 64 70 81
III 56 61 63 72
IV 71 82 83 96
2000 2001 2002 2003
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Find the linear regression equation of percentage of worms (Y) on size of the crop (X) based on the following seven observations.
Size of the crop (X) 16 15 11 27 39 22 20
Percentage of worms (Y) 24 25 34 40 35 20 23
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Number of deaths recorded in various age groups in two areas and the population size in each age group are given in the following table. Population No. of Deaths Population No. of Deaths 0-10 3000 55 7500 300 10-25 4500 30 6000 50 25-45 6000 40 8000 40 45 and over 1000 15 2000 64 Area-I Area-II Age Group (in years) Find the crude death rates and age specific death rates of Area I and Area II.
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A survey was conducted with 500 female students of which 60% were intelligent, 40% had uneducated fathers, while 30% of the not intelligent female students had educated fathers. Test the hypothesis that the education of fathers and intelligence of female students are independent.
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Construct the price indices from the following data by applying (1) Laspeyre's method (2) Paasche's method and (3) Fisher ideal number by taking 2010 as the base year.
Price (`) Quantity Price (`) Quantity
A 15 15 22 12
B 20 5 27 4
C 4 10 7 5
2010 2011
Commodity
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Construct the price indices from the following data by applying (1) Laspeyre's method (2) Paasche's method and (3) Fisher ideal number by taking 2010 as the base year.
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Following data gives the yield of wheat in India for the years 2010-11 to 2019-20. Find the trend values using 3-year moving averages.
Year 2010-11 2011-12 2012-13 2013-14 2014-15 2015-16 2016-17 2017-18 2018-19 2019-20
Yield (kg/hectare)
626 734 535 727 652 715 756 860 909 798
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