Problem 45 - Basic Statistics for Business and Economics with Student

Question

The publisher of Celebrity Living claims that the mean sales for personality magazines that feature people such as Angelina Jolie or Paris Hilton are 1.5 million copies per week. A sample of 10 comparable titles shows a mean weekly sales last week of 1.3 million copies with a standard deviation of 0.9 million copies. Does this data contradict the publisher's claim? Use the 0.01 significance level.

Step-by-Step Solution

Answer

The data does not contradict the publisher's claim.

Step 1: State the Hypotheses

We start by setting up the null and alternative hypotheses. The null hypothesis is that the mean sales for the magazines are equal to the claimed 1.5 million copies per week: \( H_0: \mu = 1.5 \text{ million} \). The alternative hypothesis is that the mean sales are not equal to the claimed amount: \( H_1: \mu eq 1.5 \text{ million} \).

Step 2: Determine the Significance Level

The significance level \( \alpha \) is given as 0.01, which means we want to be 99% confident in our results.

Step 3: Calculate the Test Statistic

The test statistic for a t-test is calculated using the formula:\[ t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \]where \( \bar{x} = 1.3 \text{ million} \), \( \mu = 1.5 \text{ million} \), \( s = 0.9 \text{ million} \), and \( n = 10 \). Substitute the values to get:\[ t = \frac{1.3 - 1.5}{\frac{0.9}{\sqrt{10}}} \approx -0.703 \]

Step 4: Determine Degrees of Freedom

The degrees of freedom for this t-test is \( n - 1 = 10 - 1 = 9 \).

Step 5: Find the Critical t-value

Using a t-distribution table or a calculator for a two-tailed test at a significance level of 0.01 with 9 degrees of freedom, we find the critical t-value to be approximately \( \pm 3.250 \).

Step 6: Compare Test Statistic to Critical Value

Compare the calculated t-value (\(-0.703\)) to the critical t-values (\(\pm 3.250\)). Since \(-0.703\) is within the range \(-3.250 < t < 3.250\), we fail to reject the null hypothesis.

Step 7: Draw a Conclusion

Since the test statistic does not fall outside the critical values, we do not have sufficient evidence to reject the publisher's claim that the mean sales are 1.5 million copies per week.

Key Concepts

t-test

The t-test is a statistical method used to determine if there is a significant difference between the means of two groups. In simpler terms, it helps us understand if variations in sample data can be explained by chance or reflect a genuine difference compared to the expected value. For the exercise with the magazine sales, the t-test is used to see if the observed mean sales differ significantly from the claimed mean of 1.5 million.

The formula for the t-test involves the sample mean, the expected mean, the sample standard deviation, and the number of observations. It's important because it considers variability and the number of data points, which affects the accuracy of results. The t-test statistic is compared against critical t-values to make decisions about hypotheses.

significance level

The significance level, denoted by \( \alpha \), is the threshold we set to determine when to reject the null hypothesis. It essentially represents our tolerance for the risk of making an incorrect decision, meaning we might mistake random variations for real effects.

For instance, in our magazine sales example, a significance level of 0.01 implies we're willing to be wrong 1% of the time. Most commonly, significance levels are set at 0.05 or 0.01 for formal studies. Choosing this level comes down to how much certainty the situation demands.

A lower significance level demands stronger evidence to reject the null hypothesis, ensuring higher certainty that findings aren't due to random chance.

critical t-value

The critical t-value is a key part of hypothesis testing in statistics. It acts as a boundary marker at which we decide to reject or fail to reject the null hypothesis. This value is determined by the sample size, the significance level, and whether the test is one-tailed or two-tailed.

In our two-tailed magazine analysis, we used a significance level of 0.01 and 9 degrees of freedom. The critical t-values were approximately \( \pm 3.250 \). These values bound the acceptance region for the test.

If the calculated t-value falls beyond these critical values, it indicates that the observed data deviate significantly from what would be expected under the null hypothesis. When the data falls within this range, we conclude there isn't enough evidence to challenge the claim.

degrees of freedom

Degrees of freedom refer to the number of values in a calculation that can vary independently. It's a crucial element in calculating the t-test statistic and ultimately impacts the critical t-value.

In the context of our problem, the degrees of freedom is computed as \( n - 1 \), where \( n \) is the sample size. Therefore, with 10 magazine sales samples, we have 9 degrees of freedom. This value helps to determine the critical t-values from a t-distribution table or calculator.

Degrees of freedom influence the shape of the t-distribution, affecting how confident you can be in your results. Always consider it when performing hypothesis testing because it dictates the precision of your conclusion.