Problem 26 - Basic Statistics for Business and Economics with Student

Question

A coffee manufacturer is interested in whether the mean daily consumption of regular-coffee drinkers is less than that of decaffeinated-coffee drinkers. Assume the population standard deviation for those drinking regular coffee is 1.20 cups per day and 1.36 cups per day for those drinking decaffeinated coffee. A random sample of 50 regular-coffee drinkers showed a mean of 4.35 cups per day. A sample of 40 decaffeinated-coffee drinkers showed a mean of 5.84 cups per day. Use the .01 significance level. Compute the \(p\) -value.

Step-by-Step Solution

Answer

The p-value is virtually 0, leading us to reject the null hypothesis.

Step 1: State the Hypotheses

Null Hypothesis, \(H_0\): The mean daily consumption of regular coffee drinkers is equal to that of decaffeinated coffee drinkers. Alternative Hypothesis, \(H_1\): The mean daily consumption of regular coffee drinkers is less than that of decaffeinated coffee drinkers.

Step 2: Identify the Means and Standard Deviations

Regular coffee: \( \mu_R = 4.35\), \( \sigma_R = 1.20\), \(n_R = 50\). Decaffeinated coffee: \( \mu_D = 5.84\), \( \sigma_D = 1.36\), \(n_D = 40\).

Step 3: Calculate the Standard Error of the Difference

The standard error (SE) is given by \[SE = \sqrt{\frac{\sigma_R^2}{n_R} + \frac{\sigma_D^2}{n_D}}\]Substitute the values:\[SE = \sqrt{\frac{1.20^2}{50} + \frac{1.36^2}{40}} = \sqrt{0.0288 + 0.04624} \approx 0.285\]

Step 4: Compute the Test Statistic

The test statistic \(Z\) is calculated as: \[Z = \frac{\mu_R - \mu_D}{SE}\]Substitute the means and SE:\[Z = \frac{4.35 - 5.84}{0.285} \approx -5.23\]

Step 5: Determine the p-value

Using a standard normal distribution table, find the \(p\)-value associated with \(Z = -5.23\). The \(p\)-value is virtually zero (far below the 0.01 significance level).

Step 6: Make a Decision

Since the \(p\)-value is less than the significance level of 0.01, reject the null hypothesis. This suggests that the mean daily consumption of regular coffee drinkers is significantly less than that of decaffeinated coffee drinkers.

Key Concepts

Population Standard Deviation

The population standard deviation is a measure that quantifies the amount of variation or dispersion in a set of population data. It's an essential component in hypothesis testing as it gives insights into the variability within groups. For instance, in the coffee consumption example, the population standard deviation for regular coffee is 1.20 cups per day, while for decaffeinated coffee, it is 1.36 cups per day.

• This implies that the daily coffee consumption among regular drinkers tends to be closer to its average than it is among decaffeinated drinkers.
• Lower standard deviation signifies less variability, meaning most values hover near the mean.
• Higher standard deviation indicates that the data points spread out over a wider range of values.

Understanding the population standard deviation is crucial as it helps in calculating other statistical measures, such as standard error and test statistics, ultimately contributing to the interpretation of hypothesis tests.

p-value

The p-value is a key concept in hypothesis testing. It quantifies the probability of observing a test statistic at least as extreme as the one actually observed, assuming that the null hypothesis is true. In simpler terms, it's the probability that the results of your sample data occur by random chance.

• A low p-value (typically less than 0.05) suggests that the observed data are unlikely under the null hypothesis, leading us to reject the null hypothesis.
• In the coffee consumption exercise, the calculated p-value is virtually zero, far below the threshold of 0.01.
• This indicates strong evidence against the null hypothesis, supporting the claim that mean consumption differs between regular and decaf drinkers.

Thus, the p-value serves as a "measure of evidence" against the null hypothesis, allowing us to understand how likely our results are due to random variation alone.

Significance Level

The significance level, denoted by alpha (α), is the probability of rejecting the null hypothesis when it is actually true. It represents the threshold or cut-off for deciding whether a p-value indicates a statistically significant result.

• In hypothesis testing, this is often set at 0.05, but in our case, it is 0.01, indicating stricter criteria for rejecting the null hypothesis.
• This means that we only accept a 1% risk of concluding that a difference exists when there is none.
• If the p-value is less than the significance level, we reject the null hypothesis in favor of the alternative hypothesis.

Selecting a significance level involves a balance between Type I error (false positive) and Type II error (false negative). In rigorous fields, lower significance levels are preferred as they reduce the probability of falsely claiming a discovery.

Standard Error

The standard error (SE) is a measure of the statistical accuracy of an estimate, indicating how much sample means can be expected to differ from the true population mean. It is crucial in the context of hypothesis testing as it helps in calculating test statistics like the Z-score.

• The formula for standard error in comparing two means is \[SE = \sqrt{\frac{\sigma_R^2}{n_R} + \frac{\sigma_D^2}{n_D}}\]
• In our exercise, the calculated SE was approximately 0.285, reflecting the combined variability of both samples.
• Standard error decreases as sample size increases, making estimates more precise.

Standard error plays a pivotal role as it allows us to quantify how variable our sample means are and determine the reliability of our difference between sample means through the test statistic.