Problem 25 - Basic Statistics for Business and Economics with Student

Question

Chicken Delight claims that 90 percent of its orders are delivered within 10 minutes of the time the order is placed. A sample of 100 orders revealed that 82 were delivered within the promised time. At the .10 significance level, can we conclude that less than 90 percent of the orders are delivered in less than 10 minutes?

Step-by-Step Solution

Answer

Yes, less than 90% of orders are delivered within 10 minutes.

Step 1: Define Null and Alternative Hypotheses

The first step in hypothesis testing is to establish the null and alternative hypotheses. In this case, we are testing if less than 90% of the orders are delivered on time.\[H_0: p = 0.90 \text{ (Null hypothesis: the true proportion is 90%)}\]\[H_a: p < 0.90 \text{ (Alternative hypothesis: the true proportion is less than 90%)}\]

Step 2: Choose the Significance Level

The significance level \( \alpha \) is given as 0.10. This means we are willing to accept a 10% chance of rejecting the null hypothesis if it is actually true.

Step 3: Calculate the Test Statistic

Use the formula for the test statistic for the proportion based on the normal distribution.\[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\]where \( \hat{p} = \frac{82}{100} = 0.82 \) is the sample proportion, \( p_0 = 0.90 \) is the hypothesized proportion, and \( n = 100 \) is the sample size.\[z = \frac{0.82 - 0.90}{\sqrt{\frac{0.90(1-0.90)}{100}}} = \frac{-0.08}{0.03} \approx -2.67\]

Step 4: Determine the Critical Value and Decision Rule

For a one-tailed test with \( \alpha = 0.10 \), we find the critical z-value using a z-table or calculator. The critical value for \( \alpha = 0.10 \) is \( z_{0.10} = -1.28 \). The decision rule is: reject the null hypothesis if the calculated z-value is less than the critical value.

Step 5: Compare Test Statistic and Critical Value

The calculated z-value is \(-2.67\), which is less than the critical value of \(-1.28\). This means that the test statistic falls into the rejection region.

Step 6: Conclude the Test

Since the test statistic \(-2.67\) is less than the critical value \(-1.28\), we reject the null hypothesis and conclude that there is sufficient evidence to suggest that less than 90% of Chicken Delight's orders are delivered within 10 minutes.

Key Concepts

Null Hypothesis

In the world of statistics, the null hypothesis serves as a starting assumption for our analysis. It's like a default position that says, "Nothing unusual is happening." For our Chicken Delight scenario, the null hypothesis (\(H_0\)) suggests that the true proportion of orders delivered within 10 minutes is precisely 90%. We express this mathematically as:- \(H_0: p = 0.90\)Here, "p" represents the true proportion we are interested in. The null hypothesis is what we test against evidence collected from the sample, aiming to provide a clear foundation for comparison.Understanding the null hypothesis is crucial because any deviation from this can indicate a significant finding. When we assume the null hypothesis to be true, we are implying that any observed changes or deviations are just results of chance — unless proven otherwise. This concept of 'default thinking' helps provide a standard from which we measure and test our hypothesis.

Alternative Hypothesis

Contrary to the null hypothesis, the alternative hypothesis proposes that something different is happening. It is like saying, "There could be a change, and we need to investigate it further." For Chicken Delight, the alternative hypothesis (\(H_a\)) is that less than 90% of their orders are delivered within 10 minutes.In a mathematical form:- \(H_a: p < 0.90\)The alternative hypothesis is what we hope to support through evidence gathered from the sample data. It suggests the presence of a systematic effect or change that deviates from what the null hypothesis predicts.An important aspect of the alternative hypothesis is its direction. In our case, we are looking for evidence of a decrease (less than 90%), hence the direction is "less than." The alternative hypothesis guides our testing process, helping us focus on detecting specific changes or differences and eventually leading us toward a conclusion based on statistical evidence.

Significance Level

The significance level, often represented by \( \alpha \), sets the boundary for how much risk we are willing to take to make an error in our decision-making process. For Chicken Delight, the significance level is set at 0.10. This means we accept a 10% risk of incorrectly rejecting the null hypothesis when it is actually true.Here's why this is important:

• A higher significance level (like 0.10) is more lenient, meaning we have a greater tolerance for risk and are more open to claiming an effect (like a change in delivery performance).
• A lower significance level (like 0.01 or 0.05) would be stricter, requiring stronger evidence before concluding that a true effect exists.

The significance level is key in hypothesis testing. It helps define the criterion for the decision rule, which tells us whether we should reject the null hypothesis. In our case, we compare the test statistic to the critical value related to this selected level to decide if the evidence is substantial enough to go against the null hypothesis. Understanding the significance level helps us make informed and appropriately cautious decisions in statistical testing.