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Web business You want to compare the daily sales for two different designs of Web pages for your Internet business. You assign the next 60 days to either Design A or Design B, 30 days to each. (a) Describe how you would assign the days for Design A and Design B using the partial line of random digits provided below. Then use your plan to select the first three days for using Design A. Show your method clearly on your paper. $$24005 \qquad 52114 \qquad 26224 \qquad 39078$$ (b) Would you use a one-sided or a two-sided significance test for this problem? Explain your choice. Then set up appropriate hypotheses. (c) If you plan to use Table B to calculate the P-value, what are the degrees of freedom? (d) The t statistic for comparing the mean sales is 2.06. Using Table B, what P-value would you report? What would you conclude?

Step by step solution

01

Assigning Days Using Random Digits

To assign 30 days to each design, read the random digits in pairs. Assign 'Design A' to even pairs (00-98 where both digits are even) and 'Design B' to odd pairs. Starting from the left, '24' (Design A), '00' (Design A), and '22' (Design A) are the first three pairs, so the first three days are assigned to Design A.

02

Choosing the Significance Test Type

A two-sided significance test is appropriate because you are interested in comparing the sales performance of both designs without specifically focusing on which design might result in higher or lower sales. This means you need to determine if there is any difference, not just an increase or decrease.

03

Setting Up Hypotheses

The null hypothesis (H0) is that there is no difference in the mean sales for Design A and Design B: \( H_0: \mu_A = \mu_B \). The alternative hypothesis (H1) is that there is a difference: \( H_1: \mu_A eq \mu_B \).

04

Determining Degrees of Freedom

Since each design has 30 days of data, the degrees of freedom for the P-value calculation in a two-sample t-test is \( n_A + n_B - 2 = 30 + 30 - 2 = 58 \).

05

Calculating the P-value and Drawing Conclusions

With a t statistic of 2.06 and 58 degrees of freedom, use Table B to find the P-value. This P-value is slightly less than 0.05, indicating a statistically significant difference at the 5% level. Thus, we reject the null hypothesis, concluding there is a significant difference between the sales from the two designs.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Random Assignment

Random assignment is a fundamental method in experimental research used to ensure each participant or, in this scenario, each day, has an equal chance of being assigned to either Design A or Design B. This technique minimizes biases and external influences, giving more reliable results. Let's break down how this was applied in the exercise.

- **Using Random Digits:** The exercise uses a line of random digits to assign 30 days to Design A and 30 days to Design B.
- **Pair Reading:** You read the numbers in pairs from the random digit line. Even pairs, such as '24', '00', and '22', are assigned to Design A. Odd pairs would belong to Design B.

This method assures that the assignment of days to each design is free of personal bias and random in nature.

Two-sided Significance Test

When comparing two different designs, such as Design A and Design B, a two-sided significance test is often chosen. This is because the goal is to find out if there is any significant difference in sales, regardless of whether one is higher or lower. Here鈥檚 why this approach is suitable for the exercise.

- **Objective of Testing:** A two-sided test checks for any deviations from the null hypothesis in either direction. It looks for any differences in mean sales, rather than just an increase or decrease.
- **Why Two-sided:** In the scenario, you're not just interested in knowing if one design performs better, but if they have different effects at all.

Hence, by using a two-sided test, you ensure thoroughness, capturing any potential difference in sales performance between the two designs.

Hypothesis Testing

Hypothesis testing in statistics allows us to make decisions or inferences about population parameters based on sample data. In this exercise, you'll perform hypothesis testing on the sales data for Design A and Design B. Let's explore the setup here:

- **Null Hypothesis (H0):** This assumes there's no difference in mean sales between Design A and Design B, formally written as \( H_0: \mu_A = \mu_B \).
- **Alternative Hypothesis (H1):** It suggests that there is a difference in the means, written as \( H_1: \mu_A eq \mu_B \).

The results of hypothesis testing will guide whether we reject the null hypothesis, implying a significant difference in sales between the two web page designs.

Degrees of Freedom

Degrees of freedom play an essential role in the context of statistical tests by indicating the number of values in a calculation that are free to vary. When comparing the means of two groups, as in this exercise, the degrees of freedom are calculated as follows:

- **Calculation:** For two groups, each with 30 days of data, the formula is \( n_A + n_B - 2 \). With \( n_A = 30 \) and \( n_B = 30 \), you have \( 58 \) degrees of freedom.

- **Why Important:** Degrees of freedom help determine the appropriate distribution to use when calculating the P-value for the t-test. It affects the shape of the t-distribution, which is needed when finding out whether observed differences in means are statistically significant.

Understanding degrees of freedom ensures you correctly apply statistical models to get valid results.