৪৯তম বিসিএস ⎯ পরিসংখ্যান [৯৮১]

The only strict requirement for a Z-test is that the true population standard deviation (σ) must be known.

If σ is unknown and must be estimated by the sample standard deviation (s), the t-distribution (which has heavier tails) is the correct distribution to use to account for the additional uncertainty in the estimate, especially with smaller samples.

While Z and t converge with very large n (B), the t-test is still technically correct. (A) and (D) are clear cases for using the t-test.

Source: Business Statistics, MK Roy.

Power is the probability of correctly rejecting a false H₀.

(A) A larger n reduces standard error, making it easier to detect an effect.

(B) A larger α makes it easier to reject H₀ (increasing power but also increasing Type I error risk).

(D) A larger effect is easier to detect.

(C) A larger σ increases the standard error, "spreading out" the sampling distribution and making it harder to distinguish a true effect from random variation, thereby decreasing power.

Goodness-of-Fit Test (One variable): Used to see if a sample distribution of a single categorical variable matches a hypothesized distribution (e.g., is the ratio of phenotypes 9:3:3:1 as genetics predicts?).

Test for Independence (Two variables): Used to see if two categorical variables are associated (e.g., is phone brand preference independent of gender?).

The power of a test increases as the standard error decreases.

In an independent test, the standard error includes all the variation between different subjects.

Pairing controls for this extraneous inter-subject variability.

By calculating a difference score for each pair (e.g., After - Before), the analysis effectively "cancels out" the baseline level unique to each subject.

The test then analyzes these differences, and the standard error becomes s_d / √n, which is often much smaller than the standard error in an independent test.

A confidence interval for the difference contains all null values (values for which we would not reject H₀).

The hypothesis H₀: μ₁ = μ₂ is equivalent to H₀: μ₁ - μ₂ = 0.

Since the value 0 is within the interval (-4.2, 1.8), we would fail to reject the null hypothesis at the α = 0.05 level.

The confidence interval and the hypothesis test are directly linked.

(A) is mitigated by the Central Limit Theorem.

(B) describes the effect we are testing for.

(D) is a judgment call, not an assumption violation.

The pooled t-test assumes σ₁² = σ₂². If this assumption is false (heteroscedasticity) and the group sizes are unequal, the test performs poorly—the actual Type I error rate can deviate significantly from the chosen α.

In this case, Welch's t-test (which does not assume equal variances) would be robust.

By convention, we assign the larger sample variance to s₁² to force the F-statistic to be ≥ 1.

This allows us to use the standard upper-tailed F-table (which only provides values > 1) for a two-tailed test by comparing F_calculated to F_critical(α/2).

Here, F_calc (3.5) is less than the critical value (3.59), so we fail to reject the null hypothesis of equal variances.

If we had not done this, we would have to find a lower-tailed critical value, which is more cumbersome.

Failing to reject H₀ means the data does not provide sufficient evidence against the assumption of equal means at the chosen α level.

It does not prove H₀ is true—other factors like low power or small samples could explain it.

Hypothesis testing provides evidence, not absolute proof.

α is the probability of rejecting H₀ when it is true (Type I error).

Setting it beforehand (commonly 0.05 or 0.01) maintains objectivity and controls the false positive rate.

Changing it post-analysis could lead to p-hacking or biased conclusions.

The p-value represents the probability of observing the data (or more extreme) assuming H₀ is true.

If p < α (e.g., α = 0.05), we reject H₀, indicating the sample means differ significantly.

We never "accept" H₀; we only fail to reject it if p ≥ α.

Source: Investopedia

The pooled t-test requires the assumption of homogeneity of variances (equal population variances).

If this is violated, the unpooled t-test (Welch's) is preferred, as it does not pool variances and adjusts degrees of freedom accordingly.

This assumption can be checked using tests like Levene's or F-test.

Source: Business Statistics, SP Gupta, MP Gupta. 

Once the t-statistic is computed using the formula t = (x̄₁ - x̄₂) / (s_p * √(1/n₁ + 1/n₂)), the next step is to evaluate it against the critical value (based on α and df) or find the p-value.

This determines if the observed difference is statistically significant.

Source: Live MCQ class.

In hypothesis testing for a pooled t-test, the null hypothesis typically assumes no difference between the population means (H₀: μ₁ = μ₂ or μ₁ - μ₂ = 0).

The alternative hypothesis (Hₐ) specifies the direction or nature of the difference (e.g., ≠, >, or <), depending on the research question.

The pooled t-test assumes that the two populations have equal variances (σ₁² = σ₂²).

By pooling the sample variances, we create a single, weighted estimate of this common variance (s_p²), which improves the precision of the test statistic compared to using separate variances, especially when sample sizes differ.

Degrees of freedom in the pooled t-test account for the loss of two degrees (one from each sample mean estimate), so df = n₁ + n₂ - 2.

This is used to reference the t-distribution for critical values or p-values, ensuring the test accounts for sample size variability.

The paired t-test (or dependent t-test) is used for matched pairs or repeated measures on the same subjects, such as pre- and post-treatment scores.

It analyzes the differences within pairs, reducing variability from individual differences and increasing test sensitivity compared to independent t-tests.

Assumptions include normally distributed differences and no outliers. 

Source: "Introductory Statistics 2e" by OpenStax.

The F-test is defined as a ratio of two sample variances. It directly tests if population variances are equal.

Significance level (α) = risk of Type I error = probability of rejecting a true H₀ (option b).

It also corresponds to 95% confidence (option c) in two-tailed tests.

For paired t-test, df = n - 1. Here, df = 20 - 1 = 19. 

    Step 01: Sp​=√56.5​≈7.51665

F = larger variance/smaller variance = 25/20 = 1.25

Always place larger variance in numerator.

A non-significant F-test indicates we fail to reject the null hypothesis, meaning there is no statistical evidence that group means differ.

It doesn’t prove all means are exactly equal—it just means differences are not significant at the chosen significance level.

In ANOVA, the F-test checks whether the variance explained by group differences (between-group variance) is significantly larger than the variance within groups (error variance).

It’s essentially testing the overall model fit.

Source: Live Mcq class 10.

The correct sequence:

1.Formulate H₀ and H₁
2.Choose significance level (α) and 3. decision rule (rejection region)
4. Collect data
5. Compute test statistic
6. Draw conclusion

The F-distribution is positively skewed, takes only non-negative values, and depends on two sets of degrees of freedom: numerator (df₁) and denominator (df₂).

The chi-square test is used in categorical data analysis—for example, to test independence of two categorical variables (e.g., gender vs. voting preference) or goodness-of-fit.

The chi-square distribution is positively skewed but approaches a normal distribution as the degrees of freedom increase. It takes only non-negative values.

Welch’s t-test is a robust alternative to the independent-samples t-test.

It is appropriate when variances are unequal (violation of homogeneity of variance) and/or when sample sizes differ.

Welch’s test calculates the test statistic similarly to Student’s t-test but uses an adjusted degrees of freedom formula (Welch–Satterthwaite equation) to handle unequal variances.

Welch’s t-test relaxes the assumption of equal variances required in the pooled (Student’s) t-test.

It’s especially useful when both sample sizes differ and variances are unequal

A small test statistic just means there is insufficient evidence to reject H₀.

It does not guarantee H₀ is “always true.”

The F-distribution is positively skewed and bounded below at 0.

It is never symmetric around zero.

Chi-square is used in tests of independence and goodness of fit.

It does not compare means (that’s t/F-test).

It can test variance indirectly, but not like an estimation method.

For normality checks, special tests (e.g., Shapiro-Wilk) are used.

Source: Live Mcq class lecture.

Welch’s t-test is designed for unequal variances.

It uses an adjusted df formula (Welch–Satterthwaite).

It works with unequal sample sizes and is valid for small/large n.

Correct decision: Fail to reject H₀.

But we never say we “accept” H₀ or assign it a probability.

P-value is not the probability H₀ is true—it measures evidence against H₀.

A paired t-test is used when observations are dependent (same subjects measured twice).

Equal sample size is not required. Equal population variance is required.

If both sample size and population variance is unequal, then we perform welch t test.

Z-test is valid if: 
Population variances are known, OR

Sample sizes are large (CLT applies).

Otherwise, a t-test is used.

This formula is the two-sample Z-test with known population variances. Here variances are unequal.

Step 6: Conclusion
There is significant evidence at 5% level that the two population means are different.

 
Final Answer: Reject H₀​. Populations have significantly different means.

Since 1.25 lies inside ±1.96, we fail to reject H₀. But we never say “accept H₀ as true.”

Two-sample Z-test is designed for independent samples. For paired data, paired t-test is used instead.

Claim is about increase, so it is one-tailed (right-tailed).