Testing Claims About Variation
In Exercises 5–16, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Assume that a simple random sample is selected from a normally distributed population.


Bank Lines The Jefferson Valley Bank once had a separate customer waiting line at each teller window, but it now has a single waiting line that feeds the teller windows as vacancies occur. The standard deviation of customer waiting times with the old multiple-line configuration was 1.8 min. Listed below is a simple random sample of waiting times (minutes) with the single waiting line. Use a 0.05 significance level to test the claim that with a single waiting line, the waiting times have a standard deviation less than 1.8 min. What improvement occurred when banks changed from multiple waiting lines to a single waiting line?


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Verified step by step guidance

1. Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis states that the standard deviation of the waiting times is equal to or greater than 1.8 minutes (H₀: σ ≥ 1.8). The alternative hypothesis states that the standard deviation of the waiting times is less than 1.8 minutes (H₁: σ < 1.8).
2. Step 2: Calculate the sample standard deviation (s) from the given data. Use the formula for the sample standard deviation: s = sqrt((Σ(xᵢ - x̄)²) / (n - 1)), where xᵢ represents each data point, x̄ is the sample mean, and n is the sample size.
3. Step 3: Compute the test statistic using the chi-square formula for variance: χ² = ((n - 1) * s²) / σ₀², where n is the sample size, s is the sample standard deviation, and σ₀ is the standard deviation under the null hypothesis (1.8 minutes).
4. Step 4: Determine the critical value or P-value. For a left-tailed test at a significance level of 0.05, use the chi-square distribution table with degrees of freedom (df = n - 1) to find the critical value or calculate the P-value corresponding to the test statistic.
5. Step 5: Compare the test statistic to the critical value or compare the P-value to the significance level (0.05). If the test statistic is less than the critical value or the P-value is less than 0.05, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Conclude whether the standard deviation of waiting times with the single line is significantly less than 1.8 minutes and discuss the improvement in waiting times.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents no effect or no difference, and the alternative hypothesis (H1), which represents the effect or difference we suspect. In this case, the null hypothesis would state that the standard deviation of waiting times with the single line is equal to 1.8 minutes, while the alternative hypothesis would claim it is less than 1.8 minutes.

P-value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. A smaller P-value indicates stronger evidence against the null hypothesis. In this scenario, if the P-value is less than the significance level of 0.05, we would reject the null hypothesis, suggesting that the new single line configuration has indeed reduced waiting times.

Standard Deviation

Standard deviation is a statistic that quantifies the amount of variation or dispersion in a set of data values. A lower standard deviation indicates that the data points tend to be closer to the mean, while a higher standard deviation indicates more spread out data. In the context of this question, comparing the standard deviation of waiting times before and after the implementation of a single line will help assess whether the change has led to a significant improvement in customer waiting times.