৪৯তম বিসিএস ⎯ পরিসংখ্যান [৯৮১]

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population.

It does not prove hypotheses true but helps in accepting or rejecting them with a given level of confidence.

The null hypothesis typically represents the status quo or no effect scenario and serves as the starting assumption in hypothesis testing.

It is tested against an alternative hypothesis.

Hypothesis testing is subject to two types of errors: Type I error (rejecting a true null hypothesis) and Type II error (failing to reject a false null hypothesis). These errors represent the limitations in decision accuracy.

Hypothesis testing measures statistical significance, not practical significance.

Practical significance refers to whether the result is meaningful in real life, which hypothesis testing does not address.

The p-value quantifies how likely it is to observe the data (or more extreme) given the null hypothesis is true. A low p-value suggests evidence against the null hypothesis.

Example of P-Value with Explanation
Suppose a company claims that the average battery life of their new smartphone is 10 hours. A tester wants to check if this claim is true and collects a sample of 30 phones, finding an average battery life of 9 hours.

The null hypothesis (H0): The average battery life is 10 hours (no difference).
The alternative hypothesis (H1): The average battery life is less than 10 hours.
The p-value tells us how likely it is to get a sample average of 9 hours or less if the true average really was 10 hours.

If the p-value is very small (for example, 0.02 or 2%), this means there's only a 2% chance of observing such a result (or more extreme) by random chance if the claim is correct.
Since 2% is less than a common significance level (such as 5%), we reject the null hypothesis and conclude that the battery life is likely less than claimed.
On the other hand, if the p-value were large (say 0.3 or 30%), it would mean the sample result is quite likely by chance, and we would fail to reject the null hypothesis, so we have no strong reason to doubt the claim.

 
Key Point
The p-value is not the probability that the null hypothesis is true; it is the probability of seeing the data you got (or more extreme) if the null hypothesis were true. A smaller p-value means stronger evidence against the null hypothesis.

Rejecting the null hypothesis means the sample data provides enough evidence to conclude that the alternative hypothesis is more likely, considering the chosen significance level.

It does not prove the null hypothesis false with absolute certainty.

A Type I error occurs when the test incorrectly rejects the null hypothesis even though it is true.

This is also known as a false positive.

The significance level (usually 0.05 or 5%) is the cutoff probability for deciding whether to reject the null hypothesis.

If the p-value is less than alpha, the null hypothesis is rejected.

In a two-tailed test, rejection regions exist on both ends of the sampling distribution.

The null hypothesis is rejected if the test statistic is either significantly smaller or larger than the expected value under the null hypothesis, beyond the critical values determined by the significance level.

The power of a test measures its ability to detect an effect when there actually is one (i.e., rejecting a false null hypothesis).

Increasing sample size or effect size generally increases power, while lowering significance level decreases power.

Power is not the same as Type I error, which is controlled by significance level.

The power of a test is the probability of rejecting a false null hypothesis.

Increasing sample size reduces standard error, increasing power.

Increasing effect size makes it easier to detect a difference, increasing power.

Increasing significance level α increases power, but also increases Type I error risk.

Increasing variability decreases power because it makes detecting effects harder.

A Type II error occurs when a false null hypothesis is not rejected, often called a false negative.

This is a Type I error, where a true null hypothesis is wrongly rejected, leading to a false positive conclusion.

Source: GeeksforGeeks

The probability of a Type I error is denoted by alpha (α) and represents the significance level of the test.

Source: Business Statistics, MP Gupta, SP Gupta 

A Type II error, or false negative, happens when the test fails to reject a false null hypothesis — for example, when a test does not detect a disease in a person who actually has it.

Source: GeeksforGeeks

A simple hypothesis specifies the population distribution completely with no unknown parameters, often a single value for the parameter.

A composite hypothesis, however, includes multiple possible values in the parameter space.

Since μ≤10 includes multiple values of μ (all values less than or equal to 10), it is a composite hypothesis.

Simple Hypothesis: A hypothesis that specifies the population parameter completely (i.e., assigns a single value).

Composite Hypothesis: Specifies a range or multiple possible values of a parameter instead of a single one.

The acceptance (or non-rejection) region is where the test statistic falls if the null hypothesis is not rejected.

Two-tailed tests are used when deviations can occur in either direction from the hypothesized value.

The power of a test is 1−β, representing the chance of detecting an actual effect when the null hypothesis is indeed false.

Source: Business statistics, Mk Roy

The significance level (α) is the probability threshold for rejecting a true null hypothesis (Type I error).

Formulating the null and alternative hypotheses is the foundational first step in any hypothesis testing procedure.

Source: Business Statistics, Md. Abdul Aziz 

ANOVA is specifically designed to compare the means across multiple groups simultaneously in hypothesis testing.

A one-tailed test is used when the alternative hypothesis predicts a change in a specific direction (either greater than or less than), making it appropriate for directional claims.

If the p-value is less than the significance level, there is sufficient evidence to reject the null hypothesis in favor of the alternative.

Source: Live MCQ class

The alternative hypothesis represents the researcher's claim or the effect they are trying to prove.

In this case, the researcher believes the drug will decrease cholesterol, so the alternative hypothesis is that the drug decreases cholesterol levels.

Option A is the null hypothesis.

When the test statistic falls in the acceptance region, it means we cannot reject the null hypothesis.

Therefore, we do not have enough evidence to support the alternative hypothesis that the mean pH is different from 7.0.

The term 'accepted' is generally avoided in favor of 'fail to reject'.

Since the calculated test statistic (z=2.15) is greater than the critical value (z=1.96), it falls into the rejection region.

This means we have statistically significant evidence to reject the null hypothesis and support the alternative hypothesis that the fertilizer increases crop yield.

A one-tailed test is used when the alternative hypothesis specifies a specific direction for the effect, such as an increase or a decrease.

In this case, the researcher is specifically testing for an increase in test scores, which makes it a one-tailed test.

Options A, B, and D all imply a two-tailed test, as they are looking for any change or difference.

The rule for hypothesis testing is to reject the null hypothesis if the p-value is less than or equal to the significance level (α).

Since 0.03≤0.05, the null hypothesis must be rejected.

This means the results are statistically significant, and there is sufficient evidence to support the alternative hypothesis.

The critical value is the threshold value on the distribution of the test statistic.

It separates the rejection region (where we would reject the null hypothesis) from the acceptance region (where we would fail to reject the null hypothesis).

The null hypothesis (H0​) contains the equal sign (=) and represents the assumption of no difference.

The alternative hypothesis (Ha​) represents the claim of the researcher.

Since the researcher wants to know if the height is different from 50 cm (and not specifically greater or less than), the alternative hypothesis must contain the 'not equal to' sign (≠).

The company's claim is that the drug works in under 30 minutes.

The alternative hypothesis represents the claim that the researcher is trying to find evidence for, which in this case is a directional claim (less than 30 minutes).

Therefore, the alternative hypothesis is Ha​:μ<30.

In a two-tailed test, the significance level (α) is split evenly between the two tails of the distribution.

Therefore, for an α of 0.01, the area in each tail is 0.01/2=0.005.

When the test statistic falls in the acceptance region, it means the observed data is not extreme enough to provide strong evidence against the null hypothesis.

The correct conclusion is to fail to reject the null hypothesis. We can never 'prove' the null hypothesis is true, so we avoid using the term 'accept'.

Since the p-value of 0.07 is greater than the significance level of 0.05, we do not have sufficient evidence to reject the null hypothesis.

This means we cannot conclude that the average time is different from 2 hours.

A simple hypothesis is a statement that provides an exact value for a population parameter.

For example, H0​:μ=100. It leaves no room for variation or a range of possible values.

The statement "the average height...is exactly 1.5 meters" represents a single, specific value for the population mean (μ=1.5).

By convention, a statement of equality is typically formulated as the null hypothesis (H0​).

The hypothesis H0​:μ≤50,000 specifies a range of possible values (μ could be 40,000, 45,000, 50,000, etc.), making it a composite hypothesis.

It's also a one-tailed hypothesis because it's directional, indicating a range on only one side of the specified value (less than or equal to).

The decision rule for hypothesis testing is: if the p-value is less than or equal to the significance level (α), you reject the null hypothesis.

Here, 0.02 is less than 0.05, so the null hypothesis (that the drug has no effect) is rejected in favor of the alternative hypothesis (that the drug has an effect).

Unlike a simple hypothesis, which states an exact value, a composite hypothesis is one that does not completely specify the parameter.

It includes inequalities, such as less than (<, less than or equal to (≤), greater than (>), or greater than or equal to (≥), thereby defining a range of possible values for the population parameter.

A test statistic is a standardized measure (like Z, t, χ², F) used to decide whether to reject H0.

Option A describes significance level (α).

Option C refers to margin of error.

Option D confuses the test statistic with the p-value (p-value is derived from the test statistic).

For small n, the t-distribution has heavier tails than normal.

As n→∞, the t-distribution approaches the standard normal.

Hence, for large sample sizes, the Z and t test statistics are numerically close.

Not all test statistics are standard normal.

Z follows normal when variance is known.

t follows Student’s t when variance is unknown.

χ² follows Chi-square for variance testing.

F follows F-distribution for variance ratios.

Therefore, option C is false.

A one-sample t-test uses the sample variance s², which has n−1 degrees of freedom.

α (significance level) = probability of committing Type I error= risk of rejecting H₀ when it is true 

A) is incorrect because the p-value is not the probability that the null hypothesis is true. Frequentist statistics treats the null hypothesis as either true or false; it does not assign it a probability.

B) is correct. This is the textbook definition of a p-value. It is a conditional probability: assuming H₀ is true, it measures the probability of seeing the observed data (or data more extreme) due to random chance alone.

C) is incorrect for the same reason as A. The p-value does not directly tell us the probability of the alternative hypothesis being true.

D) is incorrect because it describes a z-score, not a p-value. A z-score measures the distance in terms of standard errors. The p-value is the probability (area under the curve) associated with that z-score and beyond.

A Type II error (β) is failing to reject a false null hypothesis. Its probability is influenced by several factors.

A) Increasing the sample size (n): This is correct. A larger sample size decreases the standard error of the mean (σ/√n). This makes the sampling distribution narrower, increasing the power of the test (1-β). With a more precise estimate, it's easier to detect a true effect, thus reducing β.

B) Decreasing the sample size (n): This is incorrect. A smaller *n* increases the standard error, making the distribution wider. This makes it harder to detect a true effect, thereby increasing β, not decreasing it.

C) Increasing the true population mean μ: This is correct. β is not calculated in a vacuum; it depends on a specific alternative value. If the true μ is much larger than 50 (e.g., μ = 60), the alternative distribution is far from the null distribution. The overlap between the two distributions decreases, meaning the probability of incorrectly failing to reject H₀ (β) also decreases.

Since both A and C are correct actions that decrease β, the correct answer is D.