Statistics: SSC CGL Practice Questions

Ans: A

Explanation: We need to find the probability of rejecting H0 (p=0.4) when H1 (p=0.6) is true. H0 is rejected if we get 5 or 6 heads. Since H1 is true, we use p=0.6 to calculate this probability.

* Probability of 5 heads: P(X=5) = (6 choose 5) * (0.6)^5 * (0.4)^1 = 6 * 0.07776 * 0.4 = 0.186624
* Probability of 6 heads: P(X=6) = (6 choose 6) * (0.6)^6 * (0.4)^0 = 1 * 0.046656 * 1 = 0.046656

The probability of rejecting H0 (5 or 6 heads) when H1 is true is the sum of these probabilities: 0.186624 + 0.046656 = 0.23328.

Correct Option: A

Ans: C

Explanation:This is a problem involving the sampling distribution of a proportion. We are given the population proportion (p = 0.35), the sample size (n = 600), and we want to find the probability that the sample proportion is greater than 0.40. We need to calculate the z-score and use it to find the probability.

1. **Calculate the standard error:** The standard error of the sample proportion is given by sqrt(p * (1-p) / n). In this case, it’s sqrt(0.35 * 0.65 / 600).
2. **Calculate the z-score:** The z-score is calculated as (sample proportion – population proportion) / standard error. In this case, it’s (0.40 – 0.35) / sqrt(0.35 * 0.65 / 600).
3. **Find the probability:** We want the probability that the sample proportion is *greater* than 0.40, which means we want P(z > z-score).

The correct formula represents this correctly.

Ans: D

Explanation: To find the real salary, we need to adjust the nominal salary for inflation using the CPI. The formula is: Real Salary = (Nominal Salary / CPI in the given year) * CPI in the base year.

In this case:
* Nominal Salary (2018) = Rs 20,000
* CPI in 2018 = 300
* CPI in Base Year (2010) = 100

Real Salary = (20,000 / 300) * 100 = 6666.67

Ans: B

Explanation: The straight-line trend equation is y = a + bx, where y is the sales, x is the year (coded), a is the y-intercept, and b is the slope. The least squares method provides the following formulas: b = ∑xy / ∑x² and a = (∑y)/n where n is the number of years.

Given: ∑x = 0, ∑x² = 28, ∑xy = 70. n = 7 (number of years).

Calculate ‘b’: b = 70 / 28 = 2.5
Calculate ‘∑y’: Sum of sales in $1000s = 100 + 110 + 120 + 130 + 140 + 150 + 160 = 910
Calculate ‘a’: a = 910 / 7 = 130

So, the equation is y = 130 + 2.5x

The origin is at 2020. We need to find the coded value ‘x’ for 2024.
2020 corresponds to x=0.
2024 is 4 years after 2020, so x = 4.

Substitute x = 4 into the equation:
y = 130 + 2.5 * 4
y = 130 + 10
y = 170

Ans: B

Explanation:
* **A. If group indices increase k times, so also CLI does.** This statement is generally true. If the prices of goods and services used to calculate the CLI increase proportionally across all groups, the overall CLI will increase proportionally as well. This is because CLI is a weighted average of group indices.
* **B. CLI remains unchanged if all the group indices increase by a constant amount.** This statement is false. CLI is based on percentage changes, not absolute changes. If all group indices increase by the same *amount*, the CLI *will* change.
* **C. CLI helps determining Purchasing Power of Money.** This statement is true. The CLI reflects changes in the cost of goods and services, which directly impacts how much consumers can buy with a given amount of money.
* **D. Dearness Allowance is fixed by considering CLI.** This statement is true. Dearness Allowance (DA) is often linked to the CLI to compensate employees for the increased cost of living.

Correct Option: B

Ans: C

Explanation: If X follows a standard normal distribution (mean 0, variance 1), then X^2 follows a chi-squared distribution with one degree of freedom. This is because the square of a standard normal variable is, by definition, a chi-squared distribution with one degree of freedom.

Ans: A

Explanation: In a two-way ANOVA, the total sum of squares (SST) is partitioned into sum of squares due to Factor A (SSA), sum of squares due to Factor B (SSB), sum of squares due to the interaction between A and B (SSAB), and sum of squares due to error (SSE). When you remove a factor (say, Factor B) and conduct a one-way ANOVA, you are essentially combining the variability explained by Factor B and the interaction term (if it exists) with the error term. This means the SSE in the one-way ANOVA will capture the variability previously explained by Factor B (and any interaction involving it), along with the original SSE. Therefore, the new SSE (in the one-way ANOVA) will increase.

Correct Option: A

Ans: B

Explanation: In a two-way ANOVA, the degrees of freedom for each factor is calculated as the number of levels for that factor minus 1. Since factor B has ‘n’ levels, its degrees of freedom is n – 1.

Ans: B

Explanation:
1. **Degrees of freedom:**
* Degrees of freedom between groups (df_between) = number of groups – 1 = 3 – 1 = 2
* Degrees of freedom within groups (df_within) = total number of data points – number of groups = (3*3) – 3 = 9 – 3 = 6
* Degrees of freedom total (df_total) = total number of data points – 1 = 9 – 1 = 8

2. **Relationship between sums of squares:**
* Total sum of squares (SS_total) = Sum of squares between groups (SS_between) + Sum of squares within groups (SS_within)
* 18 = SS_between + SS_within

3. **F-statistic formula:**
* F = (Mean Square Between Groups) / (Mean Square Within Groups)
* F = (MS_between) / (MS_within)
* MS_between = SS_between / df_between
* MS_within = SS_within / df_within
* Therefore, F = (SS_between / df_between) / (SS_within / df_within)

4. **Rearranging the F-statistic to find SS_between**
* F = 1.5 = (MS_between) / (MS_within)
* 1. 5 = (SS_between / 2) / (SS_within / 6)
* 1. 5 = (SS_between * 6) / (SS_within * 2)
* 1. 5 = (SS_between * 3) / SS_within
* SS_within = 3 * SS_between / 1.5 = 2 * SS_between

5. **Solving for SS_between**
* SS_total = SS_between + SS_within
* 18 = SS_between + 2 * SS_between
* 18 = 3 * SS_between
* SS_between = 6

6. **Calculate MS_between**
* MS_between = SS_between / df_between
* MS_between = 6 / 2
* MS_between = 3

Correct Option: B

Ans: A

Explanation: A Type II error occurs when we fail to reject a null hypothesis that is actually false. This means we accept the null hypothesis when we shouldn’t have.

* **A. Rejecting a true null hypothesis:** This describes a Type I error.
* **B. Incorrectly accepting the alternative hypothesis:** This is the correct outcome if you correctly rejected the null hypothesis. It is NOT a consequence of a Type II error.
* **C. Rejecting the null hypothesis when it is true:** This describes a Type I error.
* **D. Failing to reject a false null hypothesis:** This is the definition of a Type II error.

Ans: A

Explanation: In the linear regression equation Y = mx + c, ‘m’ represents the slope. The slope (m) indicates how much Y changes for every one-unit increase in X. ‘c’ is the y-intercept.

Ans: C

Explanation: Moving averages are used to smooth out short-term fluctuations in a time series. They average the data over a specified period. This averaging helps to dampen the random, unpredictable fluctuations (irregular variation) and highlight the underlying patterns. By doing so, they also help in identifying the trend and cyclical components. The primary purpose is to reduce the impact of the irregular component. Although they help in identifying other components as well, their main purpose is smoothing the irregular variation.

Ans: B

Explanation: Cyclical variations in time series data refer to fluctuations that repeat over a longer period, typically years, as opposed to short-term, seasonal, or irregular fluctuations. The word “cyclical” itself implies a cycle, which by definition takes time to complete. Therefore the correct answer must be the one that indicates a long period.

Ans: A

Explanation: The Paasche’s Index is a price index that uses current period quantities as weights. It generally satisfies the Factor Reversal Test (which involves multiplying price and quantity index) but does not satisfy the Time Reversal Test (which involves interchanging base and current periods).

Ans: D

Explanation: The simple aggregative method involves summing the costs for each month and then calculating the percentage change.
Month 1 Total Cost: 150 + 200 + 250 + 300 = 900
Month 2 Total Cost: 160 + 180 + 270 + 320 = 930
Change in Cost: 930 – 900 = 30
Percentage Change: (Change in Cost / Month 1 Total Cost) * 100 = (30 / 900) * 100 = 3.33% (approximately)

Since we calculated approximate percentage increase, so the closest answer in the given options is
(30/900)*100 = 3.33% which is nearly equal to 2% (the options contains 2% and 5%)
Therefore, the closest change is the option D which says, Net Increase of 2%. However, there is some degree of approximation.

Ans: C

Explanation: The “test of commensurability” in index numbers means that the index is independent of the units used to measure the items. Because the index is a ratio or percentage, the units cancel each other out.
Correct Option: C