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Magazine Chapter 9: Problem 16
Harper's Index reported that \(80 \%\) of all supermarket prices end in the digit 9 or \(5 .\) Suppose you check a random sample of 115 items in a supermarket and find that 88 have prices that end in 9 or \(5 .\) Does this indicate that less than \(80 \%\) of the prices in the store end in the digits 9 or 5 ? Use \(\alpha=0.05\).
Short Answer
Expert verified
With \( \alpha = 0.05 \), there is not enough evidence to conclude that less than 80% of prices end in 9 or 5.
Step by step solution
01
Define hypotheses
We set up our null hypothesis, denoted as \( H_0 \), and the alternative hypothesis, denoted as \( H_1 \). \( H_0: p = 0.80 \) indicates that the proportion of prices ending in 9 or 5 is 80%, while \( H_1: p < 0.80 \) suggests that it is less than 80%.
02
Set significance level
The significance level \( \alpha \) is given as 0.05. This level is used to determine the threshold for rejecting the null hypothesis.
03
Calculate sample proportion
The sample proportion \( \hat{p} \) is calculated as the number of items with prices ending in 9 or 5 divided by the total sample size: \( \hat{p} = \frac{88}{115} \approx 0.7652 \).
04
Compute standard error
The standard error of the sample proportion \( \hat{p} \) is given by \( \sqrt{\frac{p(1-p)}{n}} \), where \( p = 0.80 \) and \( n = 115 \). Substitute these values to find the standard error: \( \text{SE} = \sqrt{\frac{0.80 \times 0.20}{115}} \).
05
Calculate the test statistic
The test statistic \( z \) is calculated using the formula \( z = \frac{\hat{p} - p}{\text{SE}} \). Substitute \( \hat{p} \), \( p = 0.80 \), and \( \text{SE} \) from the previous steps to compute \( z \).
06
Make a decision
Compare the computed \( z \)-value with the critical \( z \)-value for \( \alpha = 0.05 \) from the standard normal distribution. Since this is a one-tailed test, find the critical \( z \)-value that corresponds to \( \alpha = 0.05 \). If the computed \( z \)-value is smaller than the critical \( z \)-value, reject \( H_0 \).
07
Conclusion
Based on the comparison, if the \( z \)-value indicates rejection of the null hypothesis, conclude there is enough evidence to say that less than 80% of the prices end in 9 or 5. Otherwise, conclude that there is not enough evidence to support the alternative hypothesis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Null Hypothesis
Hypothesis testing is a method used to make decisions based on data analysis. One of the first and crucial steps in hypothesis testing is formulating the null hypothesis, denoted as \( H_0 \). The null hypothesis represents a statement of no effect or no difference. It is essentially the default or initial assumption that there is nothing unusual or extraordinary in the data.
In the context of the exercise, the null hypothesis is \( H_0: p = 0.80 \). This means we assume that 80% of the supermarket prices end with the digit 9 or 5. We start with this hypothesis because it is conventional to believe that existing or previous conditions still hold true unless there is sufficient evidence to suggest otherwise.
The null hypothesis is what we attempt to find evidence against. We do not aim to confirm or prove it true. Rather, hypothesis testing involves determining whether there is enough evidence from the sample data to reject \( H_0 \). If the null hypothesis cannot be rejected, it means that the evidence is not strong enough to conclude that the proportion of prices ending in 9 or 5 is different from 80%.
In the context of the exercise, the null hypothesis is \( H_0: p = 0.80 \). This means we assume that 80% of the supermarket prices end with the digit 9 or 5. We start with this hypothesis because it is conventional to believe that existing or previous conditions still hold true unless there is sufficient evidence to suggest otherwise.
The null hypothesis is what we attempt to find evidence against. We do not aim to confirm or prove it true. Rather, hypothesis testing involves determining whether there is enough evidence from the sample data to reject \( H_0 \). If the null hypothesis cannot be rejected, it means that the evidence is not strong enough to conclude that the proportion of prices ending in 9 or 5 is different from 80%.
Alternative Hypothesis
In contrast to the null hypothesis, the alternative hypothesis, denoted as \( H_1 \), represents a statement that indicates the potential effect or difference in the population. When formulating the alternative hypothesis, we specify what we believe might be true if the null hypothesis is not.
For the given exercise, the alternative hypothesis is \( H_1: p < 0.80 \). This suggests that fewer than 80% of the prices end with the digit 9 or 5. The direction of the inequality in the alternative hypothesis (less than, in this case) indicates the nature of the test - a one-tailed test.
The alternative hypothesis is key because it guides us on what to look for in the sample data. If we find that our data significantly leans toward \( H_1 \), then we have grounds to reject the null hypothesis in favor of the alternative hypothesis. Thus, when conducting the hypothesis test, the goal is to assess whether the evidence provided by the data is strong enough to support \( H_1 \).
For the given exercise, the alternative hypothesis is \( H_1: p < 0.80 \). This suggests that fewer than 80% of the prices end with the digit 9 or 5. The direction of the inequality in the alternative hypothesis (less than, in this case) indicates the nature of the test - a one-tailed test.
The alternative hypothesis is key because it guides us on what to look for in the sample data. If we find that our data significantly leans toward \( H_1 \), then we have grounds to reject the null hypothesis in favor of the alternative hypothesis. Thus, when conducting the hypothesis test, the goal is to assess whether the evidence provided by the data is strong enough to support \( H_1 \).
Significance Level
The significance level, denoted as \( \alpha \), is a threshold that helps us determine whether to reject the null hypothesis. Typically, \( \alpha \) is set at 0.05, 0.01, or 0.10. In hypothesis testing, it's a critical value that represents the probability of rejecting the null hypothesis when it is actually true (a Type I error).
In the exercise at hand, the significance level is \( \alpha = 0.05 \). This means there's a 5% risk that we reject \( H_0 \) while it's true. Lower significance levels suggest more stringent criteria for evidence against the null hypothesis.
Choosing \( \alpha = 0.05 \) reflects a balance between being cautious (not too easily rejecting \( H_0 \)) and being receptive to potential discoveries. This significance level is used to determine the critical value from a statistical distribution, which is then compared to the calculated test statistic. If the test statistic falls into the rejection region beyond the critical value, then \( H_0 \) is rejected in favor of \( H_1 \).
In the exercise at hand, the significance level is \( \alpha = 0.05 \). This means there's a 5% risk that we reject \( H_0 \) while it's true. Lower significance levels suggest more stringent criteria for evidence against the null hypothesis.
Choosing \( \alpha = 0.05 \) reflects a balance between being cautious (not too easily rejecting \( H_0 \)) and being receptive to potential discoveries. This significance level is used to determine the critical value from a statistical distribution, which is then compared to the calculated test statistic. If the test statistic falls into the rejection region beyond the critical value, then \( H_0 \) is rejected in favor of \( H_1 \).
Sample Proportion
The sample proportion \( \hat{p} \) is a key statistic used to estimate the proportion of a particular characteristic in the population, based on a sample. It is calculated by dividing the number of successes in the sample by the total sample size.
In the exercise, the sample proportion is calculated as \( \hat{p} = \frac{88}{115} \approx 0.7652 \). Here, "successes" refer to the number of items whose prices end in 9 or 5. This proportion offers a snapshot of the characteristic in the sample, which we then compare to the assumed population proportion under the null hypothesis.
The sample proportion is simple yet powerful, as it helps form the basis of computing other statistics, such as standard error and test statistic. The closer \( \hat{p} \) is to the population proportion under \( H_0 \), the less likely we are to reject the null hypothesis. Contrastingly, significant deviations from \( p = 0.80 \) suggest that the null hypothesis might not hold, thereby lending support to the alternative hypothesis.
In the exercise, the sample proportion is calculated as \( \hat{p} = \frac{88}{115} \approx 0.7652 \). Here, "successes" refer to the number of items whose prices end in 9 or 5. This proportion offers a snapshot of the characteristic in the sample, which we then compare to the assumed population proportion under the null hypothesis.
The sample proportion is simple yet powerful, as it helps form the basis of computing other statistics, such as standard error and test statistic. The closer \( \hat{p} \) is to the population proportion under \( H_0 \), the less likely we are to reject the null hypothesis. Contrastingly, significant deviations from \( p = 0.80 \) suggest that the null hypothesis might not hold, thereby lending support to the alternative hypothesis.
