Unit 12: Hypothesis Testing - Subjective Questions

A statistical hypothesis is a testable statement about a population parameter, such as its mean, proportion, or variance.

• The null hypothesis, denoted by , represents the existing condition, no effect, or no significant difference.
• The alternative hypothesis, denoted by or , contradicts and represents the effect or difference that the researcher wants to establish.

Example: Suppose a company claims that the mean life of its batteries is 500 hours. The hypotheses may be written as:

Here, rejecting would indicate that the mean battery life differs significantly from 500 hours.

The principal steps in hypothesis testing are:

1. State the hypotheses: Formulate the null hypothesis and alternative hypothesis .
2. Select the significance level: Choose , commonly or .
3. Choose an appropriate test: Select a -test, -test, chi-square test, or another test according to the data and assumptions.
4. Determine the decision rule: Identify the critical value or specify the p-value approach.
5. Calculate the test statistic: Use sample information to calculate the appropriate statistic.
6. Make the statistical decision: Reject if the statistic falls in the critical region or if ; otherwise, fail to reject .
7. State the conclusion: Interpret the decision in the context of the original business or research problem.

A simple hypothesis completely specifies the value of the population parameter and therefore identifies a single population distribution.

Example:

A composite hypothesis allows the parameter to take any of several possible values or a range of values.

Examples:

or

Thus, a simple hypothesis specifies one exact parameter value, whereas a composite hypothesis covers multiple parameter values. In practice, the null hypothesis is frequently simple, while the alternative hypothesis is often composite.

Errors can occur because a decision about a population is made using sample evidence.

• A Type I error occurs when a true null hypothesis is rejected. Its probability is denoted by :

• A Type II error occurs when a false null hypothesis is not rejected. Its probability is denoted by :

For example, rejecting a shipment that actually meets quality standards is a Type I error, whereas accepting a shipment that does not meet the standards is a Type II error. For a fixed sample size, reducing one type of error often increases the other.

The level of significance, denoted by , is the maximum probability of committing a Type I error that the researcher is willing to accept.

Common choices are:

• , representing a 5% risk of rejecting a true .
• , representing a 1% risk of rejecting a true .

The value of determines the rejection region and the critical value of the test. Under the p-value method:

• If , reject .
• If , fail to reject .

A smaller significance level demands stronger sample evidence before the null hypothesis can be rejected.

A one-tailed test is used when the alternative hypothesis specifies a direction.

• Right-tailed test:

• Left-tailed test:

A two-tailed test is used when departures in either direction are important:

In a one-tailed test, the entire rejection region lies in one tail. In a two-tailed test, it is divided equally between both tails. The choice must be based on the research question and made before examining the sample results.

Under the critical-value approach, a critical value is obtained from the sampling distribution using the chosen significance level. The calculated test statistic is then compared with this value.

• Reject if the statistic lies in the rejection region.
• Otherwise, fail to reject .

Under the p-value approach, the p-value measures how probable a result at least as extreme as the observed result would be if were true.

• Reject if .
• Fail to reject if .

Both approaches produce the same statistical decision when they use the same test and significance level. The p-value additionally indicates the strength of evidence against .

The power of a test is the probability of correctly rejecting a false null hypothesis. It is given by:

where is the probability of a Type II error.

The power is affected by:

• Sample size: A larger sample generally increases power.
• Significance level: A larger generally increases power but also raises the risk of a Type I error.
• Effect size: Larger actual differences from the null value are easier to detect.
• Population variability: Lower variability generally increases power.
• Test design: A correctly chosen one-tailed test can have greater power in the specified direction.

High power is desirable because it improves the ability of a test to identify meaningful effects.

Step 1: State the hypotheses

Step 2: Calculate the test statistic

Because the population standard deviation is known, use the -statistic:

Substituting the values:

Step 3: Apply the decision rule

For a two-tailed test at , the critical values are and .

Since , the test statistic lies in the rejection region.

Conclusion: Reject . There is sufficient evidence at the 5% significance level to conclude that the mean product weight differs from 50 grams.

Step 1: Formulate the hypotheses

This is a right-tailed test.

Step 2: Calculate the test statistic

Since the population standard deviation is unknown and , use a one-sample -test:

The degrees of freedom are:

Step 3: Make the decision

For a right-tailed test with and , the critical value is approximately .

Since , reject .

Conclusion: There is sufficient evidence at the 5% level to conclude that the average task completion time has increased beyond 30 minutes.

Both tests assess hypotheses about a population mean, but they are used under different conditions.

-test:

• Commonly used when the population standard deviation is known.
• May also be used for a sufficiently large sample under appropriate conditions.
• Uses the standard normal distribution.
• The test statistic is:

-test:

• Used when is unknown and is estimated by the sample standard deviation .
• Especially important for small samples from an approximately normal population.
• Uses Student's -distribution with degrees of freedom.
• The test statistic is:

As the sample size increases, the -distribution approaches the standard normal distribution.

Step 1: State the hypotheses

This is a left-tailed test.

Step 2: Calculate the sample proportion

Step 3: Calculate the test statistic

Step 4: Make the decision

At for a left-tailed test, the critical value is approximately .

Since , reject .

Conclusion: There is sufficient evidence at the 5% level to conclude that the customer satisfaction rate is below 60%.

A hypothesis test evaluates whether the sample provides sufficient evidence against . If the evidence is strong, is rejected. If it is not strong enough, the correct decision is to fail to reject .

Failure to reject does not prove that is true because:

• The sample may be too small to detect an actual effect.
• The test may have low statistical power.
• Sampling variation may conceal the true difference.
• Other parameter values close to the null value may also be consistent with the data.

Therefore, the statement fail to reject accurately indicates insufficient evidence against the null hypothesis without claiming that it has been proven.

Important assumptions and conditions include:

• Random sampling: The sample should be randomly selected so that it represents the population.
• Independence: Observations should be independent. When sampling without replacement, the sample is commonly required to be no more than 10% of the population.
• Normality: For a small-sample -test, the population should be approximately normal, with no severe outliers.
• Large-sample condition: With a sufficiently large sample, the sampling distribution of is approximately normal by the central limit theorem.
• Measurement level: The variable should be quantitative.
• Known or unknown variance: Whether is known determines whether a -test or -test is appropriate.

Violations of these assumptions can make the test statistic, p-value, and conclusion unreliable.

In quality control, hypothesis testing helps determine whether a production process or shipment satisfies a required standard.

For example:

A sample of products may be tested for weight, strength, lifetime, or defect rate.

• A Type I error means concluding that a process is unacceptable when it actually meets the standard. This may cause unnecessary production stoppages, inspections, adjustments, and financial losses.
• A Type II error means failing to detect that a process is unacceptable. This may result in defective goods, customer complaints, warranty costs, or reputational damage.

Managers should choose the significance level and sample size after considering the relative costs of these errors.

A business may compare average sales, conversion rates, or customer awareness before and after an advertising campaign or between treatment and control groups.

For example, if is mean sales under the new campaign and is mean sales under the old campaign, the hypotheses could be:

The business would:

1. Select representative samples or conduct a randomized experiment.
2. Choose an appropriate significance level.
3. Use a test for means or proportions, depending on the outcome.
4. Calculate the test statistic and p-value.
5. Decide whether the observed improvement is statistically significant.
6. Consider practical significance, campaign cost, expected profit, and return on investment.

Statistical significance alone does not guarantee that the campaign is economically worthwhile.

Statistical significance indicates that an observed result is unlikely to have occurred by random sampling variation under . It is commonly established when:

Practical significance considers whether the size of the observed effect is large enough to be useful or economically important.

For example, a large retailer may find that a new store layout increases average customer spending by with a very small p-value. The increase is statistically significant because of the large sample, but it may not be practically significant after redesign costs are considered.

Business decisions should therefore consider:

• Effect size
• Implementation cost
• Expected revenue or savings
• Risks and operational consequences
• Confidence intervals

A sound decision requires both statistical evidence and managerial judgment.

Let denote the mean payment time under the old system and the mean under the new system. The hypotheses may be stated as:

The bank should then:

1. Design the study: Randomly assign comparable customers to the old and new reminder systems, if feasible.
2. Choose : For example, use .
3. Select the test: Use an independent two-sample -test for separate groups or a paired -test if matched observations are available.
4. Check assumptions: Assess independence, randomization, outliers, and approximate normality or adequate sample size.
5. Calculate the statistic and p-value.
6. Make a decision: Reject if the p-value does not exceed .
7. Interpret the result: Evaluate both the statistical evidence and whether the reduction in payment time justifies the system's cost.

These factors influence the standard error, test statistic, and statistical power.

• Sample size: A larger reduces the standard error. For a population mean, the standard error is or . This makes it easier to detect a real difference.
• Effect size: A larger difference between the true parameter and the null value produces a larger absolute test statistic and is more likely to be detected.
• Variability: Greater variability increases the standard error, reduces the absolute test statistic, and makes differences harder to detect.

For example, the general form of a mean test statistic is:

Thus, larger samples, larger effects, and lower variability generally increase power. Businesses should plan sample sizes before collecting data so that economically meaningful effects can be detected.

The p-value is the probability, calculated under the assumption that is true, of obtaining a test result at least as extreme as the observed result.

Given:

(a) At :

Since , reject . The result is statistically significant at the 5% level.

(b) At :

Since , fail to reject . The result is not statistically significant at the 1% level.

The p-value does not represent the probability that is true, nor does it measure the size or business importance of the effect. It measures the compatibility of the observed data with under the assumptions of the test.