Math Problem Statement

An automotive safety agency is interested in comparing the distance at which a sign can be read at night using two different headlight designs. To do so, eight randomly selected drivers were tested using both headlight designs. The distance (in yards) at which the sign could be read is given in the Excel spreadsheet. At the 1% significance level, is the average reading distance using headlight design 2 greater than that of headlight design 1?   Click on Files on the left-hand side under the quiz timer. Once you click on Files, you will see the following Excel file on the right-hand side under Instructor-attached filesQuiz6.xlsx. Download Quiz6.xlsx to your computer and then open it.   ****    Use this Excel sheet: Headlight      Use columns: B (Design 1) and C (Design 2)

       

Variable 1: Design 1Variable 2: Design 2   a.  The samples are  (Click to select)  dependent  independent  .   b.  State the null and alternative hypotheses.

     

multiple choice 2

H0: μD = 0; HAμD > 0

H0: μD = 0; HAμD ≠ 0

H0: μD = 0; HAμD < 0

  c.  Calculate the test statistic and the p-value. Round to four decimal places.          test statistic =          ** p-value =**    d.  At the 1% significance level, is the average reading distance using headlight design 2 greater than that of headlight design 1?  

multiple choice 3

Yes, since we reject H0.Design 1

Design 2

124

136

91

87

103

117

102

115

112

117

75

71

87

101

118

131

Yes, since we do not reject H0.

No, since we reject H0.

No, since we do not reject H0.

Design 1

124

91

103

102

112

75

87

118

Solution

Let's go through each part of the question:

a. The samples are dependent since the same drivers tested both headlight designs.

b. The null and alternative hypotheses are:

  • H0:μD=0H_0: \mu_D = 0 (There is no difference in reading distance between the two headlight designs).
  • HA:μD>0H_A: \mu_D > 0 (The average reading distance using headlight design 2 is greater than that of headlight design 1).

c. Calculations:

  • Test Statistic = 2.8259
  • p-value = 0.0128

d. At the 1% significance level, we do not reject H0H_0 because the p-value (0.0128) is greater than 0.01. Therefore, there is not enough evidence to conclude that the average reading distance using headlight design 2 is greater than that of headlight design 1 at the 1% significance level.

Would you like a detailed explanation of the paired t-test process or have any other questions?


Here are 5 related questions for further exploration:

  1. What factors might influence the readability of signs besides headlight design?
  2. How does a paired t-test differ from an independent t-test?
  3. Why is the significance level important in hypothesis testing?
  4. What might the results imply for future headlight design improvements?
  5. Could different environmental conditions impact the results of this test?

Tip: Always check assumptions (like normality) in statistical tests to ensure accurate results.

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Math Problem Analysis

Mathematical Concepts

Paired Sample T-Test
Hypothesis Testing
Statistical Significance
P-Value

Formulas

Paired sample t-test formula: t = (D̄ - μD) / (sD / √n)
Where D̄ is the mean of the differences, μD is the hypothesized mean difference, sD is the standard deviation of the differences, and n is the sample size.

Theorems

Central Limit Theorem (applies to the sampling distribution)
Hypothesis Testing Framework

Suitable Grade Level

College Undergraduate