1 Suppose Prior Elections In A Certain State Indicated It Is Necessar ✓ Solved

1. Suppose prior elections in a certain state indicated it is necessary for a candidate for governor to receive at least 80% of the vote in the northern part of the state to be elected. The incumbent governor is interested in assessing his chances of returning to office and plans to conduct a survey of 2,000 registered voters in the northern section of the state. Using the hypothesis-testing procedure, assess the governor’s chances of re-election. 2.

Clean cab co. offer services from Nairobi CBD to JKIA airport. The president of the company is considering two routes. One is through Jogoo road and the other through Mombasa road. He wants to study the time it takes to drive to the airport using each route and then compare the results. He collected the following sample data, which is reported in minutes in the table below. Jogoo road Mombasa road Using the 0.10 significance level, is there a difference in variation in the driving time for the two routes Facilitator: Patrick Kimaku –

Paper for above instructions

Assignment Solution: Analysis of Election Chances and Route Variability

Part 1: Election Chances Assessment

Introduction

The assessment of the incumbent governor's chances of re-election requires a hypothesis testing approach. In this context, the goal is to determine whether the governor can secure at least 80% of the vote in the northern part of the state based on a survey conducted among registered voters.

Hypothesis Formation

To conduct a hypothesis test, we define our null and alternative hypotheses:
- Null Hypothesis (H0): The proportion of voters supporting the incumbent governor, \( p \), is less than or equal to 0.80.
- Alternative Hypothesis (H1): The proportion of voters supporting the incumbent governor, \( p \), is greater than 0.80.

Data Collection

The governor plans to survey 2,000 registered voters. Let's assume from his survey responses, he receives the following results:
- Number of voters supporting the governor: 1,600
- Proportion of support: \( p̂ = \frac{1600}{2000} = 0.80 \)

Test Statistic Calculation

We will use a one-sample z-test for proportions. The test statistic can be calculated using the formula:
\[
z = \frac{p̂ - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}
\]
Where:
- \( p̂ = 0.80 \) (sample proportion)
- \( p_0 = 0.80 \) (hypothesized population proportion)
- \( n = 2000 \) (sample size)
In this case, we have:
\[
z = \frac{0.80 - 0.80}{\sqrt{\frac{0.80(1-0.80)}{2000}}}
= \frac{0}{\sqrt{\frac{0.16}{2000}}}
= \frac{0}{0.0894} = 0
\]

Critical Value and Decision Rule

To determine the critical value at a significance level of \( \alpha = 0.05 \) for a one-tailed test, we can refer to the z-table. The critical value is approximately 1.645.
Decision Rule:
- If \( z > 1.645 \), reject the null hypothesis.
- If \( z \leq 1.645 \), do not reject the null hypothesis.
Given our calculated z-value of 0, we do not reject the null hypothesis.

Conclusion

The results of the hypothesis test indicate that there is insufficient evidence to conclude that the governor receives more than 80% of the vote in the northern section of the state. Thus, based on this analysis, the incumbent's chances of re-election are uncertain.
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Part 2: Analysis of Driving Times

Introduction

The second part of the assignment involves investigating whether there is a significant difference in the variation of driving times between two routes: Jogoo Road and Mombasa Road.

Data Collection

Assuming the collected sample data for driving times is as follows (in minutes):
| Route | Sample Driving Times (minutes) |
|----------------|-------------------------------|
| Jogoo Road | 30, 32, 29, 31, 28 |
| Mombasa Road | 35, 38, 37, 36, 34 |

Hypothesis Formation

To assess the variability in driving time between the two routes, we will conduct an F-test for equality of variances.
- Null Hypothesis (H0): The variances of the driving times for both routes are equal (\( \sigma^2_1 = \sigma^2_2 \)).
- Alternative Hypothesis (H1): The variances of the driving times for both routes are not equal (\( \sigma^2_1 \neq \sigma^2_2 \)).
###Calculating Sample Variances
The sample variances for both routes can be calculated as follows:
1. Calculate the means:
- Mean of Jogoo Road (\( \bar{X}_1 \)):
\[
\bar{X}_1 = \frac{30 + 32 + 29 + 31 + 28}{5} = 30
\]
- Mean of Mombasa Road (\( \bar{X}_2 \)):
\[
\bar{X}_2 = \frac{35 + 38 + 37 + 36 + 34}{5} = 36
\]
2. Calculate the sample variances:
- Variance of Jogoo Road (\( s_1^2 \)):
\[
s_1^2 = \frac{(30-30)^2 + (32-30)^2 + (29-30)^2 + (31-30)^2 + (28-30)^2}{n-1} \approx 2
\]
- Variance of Mombasa Road (\( s_2^2 \)):
\[
s_2^2 = \frac{(35-36)^2 + (38-36)^2 + (37-36)^2 + (36-36)^2 + (34-36)^2}{n-1} \approx 2
\]

Test Statistic Calculation

We will compute the F-statistic using the formula:
\[
F = \frac{s_1^2}{s_2^2}
\]
Assuming equal variances, substituting our values:
\[
F = \frac{2}{2} = 1
\]

Critical Value Determination

To determine the critical value of F at a significance level of \( \alpha = 0.10 \), we will need to use the F-distribution table with degrees of freedom:
- \( df_1 = n_1 - 1 = 5 - 1 = 4 \)
- \( df_2 = n_2 - 1 = 5 - 1 = 4 \)
From the F-distribution table, the critical value \( F(0.10, 4, 4) \) is approximately 4.29.

Conclusion

We now compare our calculated F-statistic (1) against the critical value (4.29):
- Since \( 1 < 4.29 \) we fail to reject the null hypothesis.
This means there is no statistically significant difference in the variation of driving times between Jogoo Road and Mombasa Road at the 0.10 significance level.

Overall Conclusion

The hypothesis tests conducted reveal that the incumbent governor has not demonstrated sufficient support to promise re-election. Meanwhile, there is no significant difference in driving times between the analyzed routes, indicating that the route selection based solely on variation in travel time may not be necessary.

References

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