When $H_{1}$ is a one-sided (right) alternative hypothesis, the critical region is determined by :
a.
right tail
Correct Answer
b.
both right and left tails
c.
left tail
d.
neither right nor left tail
Step-by-Step AI Explanation
Verified Method
For a one-sided (right) alternative hypothesis, the rejection region (critical region) lies entirely in the right tail of the sampling distribution. Option (b) corresponds to a two-tailed test, (c) to a left-tailed test, and (d) is incorrect as a test must have a critical region.
Q2mcq
If the size of sample is larger than 30, the t-distribution tends to :
a.
Chi-square distribution
b.
Normal distribution
Correct Answer
c.
F-distribution
d.
Poisson distribution
Step-by-Step AI Explanation
Verified Method
According to the properties of the t-distribution, as the degrees of freedom (which depend on sample size n) increase, the t-distribution approaches the Standard Normal distribution (Z-distribution). For $n > 30$, the approximation is considered sufficiently accurate.
Q3mcq
Critical value at 5% level of significance for two-tailed large sample test is :
a.
2.58
b.
1.645
c.
1.96
Correct Answer
d.
2.33
Step-by-Step AI Explanation
Verified Method
For a two-tailed test at the 5% level of significance ($\alpha = 0.05$), the critical regions are divided into 2.5% in each tail. The Z-score corresponding to 0.025 in the tail is approximately 1.96. Thus, the critical value is 1.96.
Q4mcq
The null hypothesis in the ANOVA is that all the population means are :
a.
Unequal
b.
Equal
Correct Answer
c.
Either equal or unequal
d.
Variable
Step-by-Step AI Explanation
Verified Method
In Analysis of Variance (ANOVA), the null hypothesis ($H_0$) states that there is no significant difference between the group means, implying that all population means are equal, i.e., $\mu_1 = \mu_2 = ... = \mu_k$.
Q5mcq
The statistic $$\chi^{2}$$, with usual notations, in case of contingency table of order (m×n) is given by :
For a contingency table of order $m \times n$, the $\chi^2$ statistic is calculated by summing the squared difference between observed ($O_{ij}$) and expected ($E_{ij}$) frequencies divided by the expected frequency over all rows ($m$) and columns ($n$). The formula is $\chi^{2}=\sum_{i=1}^{m}\sum_{j=1}^{n} \frac{(O_{ij}-E_{ij})^{2}}{E_{ij}}$.
Q6mcq
ANOVA was developed by :
a.
W.S. Gosset
b.
S.D. Poisson
c.
Karl Pearson
d.
R.A. Fisher
Correct Answer
Step-by-Step AI Explanation
Verified Method
Analysis of Variance (ANOVA) was developed by Sir Ronald A. Fisher in the 1920s. He is widely recognized as the father of modern statistics and the creator of this technique.
Q7mcq
If r=11, then the correlation is called :
a.
Perfect negative correlation
b.
Perfect positive correlation
Correct Answer
c.
Positive correlation
d.
Negative correlation
Step-by-Step AI Explanation
Verified Method
The coefficient of correlation $r$ ranges from -1 to +1. If $r = 1$, it indicates a perfect positive correlation, meaning the variables move in the same direction in exact proportion. Note: The provided question states r=11, which is mathematically impossible for a correlation coefficient; assuming it is a typo for r=1, (b) is the correct definition.
Q8mcq
Correlation co-efficient is the __________ between the regression coefficients.
a.
harmonic mean
b.
arithmetic mean
c.
combined mean
d.
geometric mean
Correct Answer
Step-by-Step AI Explanation
Verified Method
The correlation coefficient $r$ is the geometric mean of the two regression coefficients $b_{xy}$ and $b_{yx}$. This is represented by the formula $r = \pm \sqrt{b_{xy} \cdot b_{yx}}$.
Q9mcq
The correlation coefficient of x and y is symmetric. Hence :
a.
r_{xy} < r_{yx}
b.
r_{xy} = r_{yx}
Correct Answer
c.
r_{xy} ≠ r_{yx}
d.
r_{xy} > r_{yx}
Step-by-Step AI Explanation
Verified Method
The correlation coefficient is a measure of the linear relationship between two variables. Mathematically, it is defined as $r_{xy} = \frac{Cov(x, y)}{\sigma_x \sigma_y}$. Since the covariance is symmetric ($Cov(x, y) = Cov(y, x)$) and the product of standard deviations is commutative, $r_{xy} = r_{yx}$. Thus, choice (b) is correct.
Q10mcq
History of a cohort can be understood from :
a.
Fertility rates
b.
Mortality rates
c.
Population growth
d.
Life table
Correct Answer
Step-by-Step AI Explanation
Verified Method
A life table (also known as a mortality table) provides a summary of the mortality and survival experience of a specific generation or cohort over time, tracking them from birth to death. Thus, (d) is correct.
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A Primary Health Centre located in a village has a record of 135 live births during a year. The number of deaths recorded in the village during that year was 300. Among them, 5 are newborn babies of age less than a year. Measure the infant mortality rate of the village.
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Three composition instructors recorded the number of spelling errors which their students made on a research paper. At 5% level of significance test whether there is significant difference in the average number of errors.
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Three composition instructors recorded the number of spelling errors which their students made on a research paper. At 5% level of significance test whether there is significant difference in the average number of errors in the three classes of students.
Instructor 1: 2 3 5 0 8
Instructor 2: 4 6 8 4 9 0 2
Instructor 3: 5 2 3 2 3 3
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Can vaccination be regarded as a preventive measure of Hepatitis B from the data given below ?
Of 1500 persons in a locality, 400 were attacked by Hepatitis B. 750 has been vaccinated. Among them only 75 were attacked.
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The mean breaking strength of cables supplied by a manufacturer is 1900 $n/m^2$ with a standard deviation of 120 $n/m^2$. 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 $n/m^2$. Can we support the claim at 1% level of significance ?
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An IQ test was conducted to 5 persons before and after they were trained. The results are given below :
Candidates: I II III IV V
IQ before training: 110 120 123 132 125
IQ after training: 120 118 125 136 121
Test whether any improvement in IQ at 1% level of significance.
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Obtain the seasonal indices for the rainfall (in mm) data in India given in the following table.
Year: 2009 2010 2011 2012
I: 38.2 38.5 55 50.5
II: 166.8 250.9 277.7 197
III: 612.6 773.1 717.8 706.1
IV: 72.2 153.1 65.8 101.1
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The following is the data regarding the size of female population in a country at reproductive age and the live births during a period.
Age Groups: 15-20, 20-25, 25-30, 30-35, 35-40, 40-45
Female Pop: 2,16,410 | 2,13,610 | 2,02,930 | 1,93,300 | 1,73,920 | 1,62,870
Live Births: 20,468 | 26,983 | 22,522 | 17,083 | 13,456 | 11,140
Calculate the general fertility rate and age specific fertility rates of the country.
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Obtain the seasonal indices for the rainfall (in mm) data in India given in the following table.
| Quarter | I | II | III | IV |
|---|---|---|---|---|
| 2009 | 38.2 | 166.8 | 612.6 | 72.2 |
| 2010 | 38.5 | 250.9 | 773.1 | 153.1 |
| 2011 | 55 | 277.7 | 717.8 | 65.8 |
| 2012 | 50.5 | 197 | 706.1 | 101.1 |
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The following is the data regarding the size of female population in a country at reproductive age and the live births during a period.
| Age Groups (Years) | Female Population | No. of Live Births |
|---|---|---|
| 15 - 20 | 2,16,410 | 20,468 |
| 20 - 25 | 2,13,610 | 26,983 |
| 25 - 30 | 2,02,930 | 22,522 |
| 30 - 35 | 1,93,300 | 17,083 |
| 35 - 40 | 1,73,920 | 13,456 |
| 40 - 45 | 1,62,870 | 11,140 |
Calculate the general fertility rate and age specific fertility rates of the country.
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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.
| Commodity | 2010 Price (`) | 2010 Quantity | 2011 Price (`) | 2011 Quantity |
|---|---|---|---|---|
| A | 15 | 15 | 22 | 12 |
| B | 20 | 5 | 27 | 4 |
| C | 4 | 10 | 7 | 5 |
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