Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
Indeed, let's provide a detailed step-by-step solution based on the given data.
### Hypothesis Testing for the Mean
We want to test if the mean miles driven (in thousands) by vehicles differs from a hypothesized mean.
Given Data:
1. Mean miles driven, [tex]\( \overline{x} = 150.7 \)[/tex]
2. Variance, [tex]\( s^2 = 4.661 \)[/tex]
3. Number of observations, [tex]\( n = 46 \)[/tex]
4. Hypothesized mean, [tex]\( \mu = 150 \)[/tex]
5. Degrees of freedom, [tex]\( df = 45 \)[/tex]
6. Test Statistic, [tex]\( t = 2.185 \)[/tex]
7. One-tail p-value, [tex]\( p_{\text{one-tail}} = 0.017 \)[/tex]
8. Two-tail p-value, [tex]\( p_{\text{two-tail}} = 0.034 \)[/tex]
9. Critical value (one-tail), [tex]\( t_{\text{critical (one-tail)}} = 1.679 \)[/tex]
10. Critical value (two-tail), [tex]\( t_{\text{critical (two-tail)}} = 2.014 \)[/tex]
11. Confidence Level, [tex]\( 95\% \)[/tex]
### Step-by-step Explanation:
#### Step 1: State the Hypotheses
We are performing a t-test to test the following hypotheses:
- Null hypothesis: [tex]\( H_0: \mu = 150 \)[/tex] (where [tex]\( \mu \)[/tex] is the population mean)
- Alternative hypothesis: [tex]\( H_a: \mu \neq 150 \)[/tex] (this is a two-tailed test)
#### Step 2: Calculate the Test Statistic
The test statistic is calculated using the formula for the t-test for a single sample:
[tex]\[ t = \frac{\overline{x} - \mu}{s / \sqrt{n}} \][/tex]
Where:
- [tex]\( \overline{x} \)[/tex] is the sample mean
- [tex]\( \mu \)[/tex] is the hypothesized population mean
- [tex]\( s \)[/tex] is the sample standard deviation (the square root of the variance)
- [tex]\( n \)[/tex] is the number of observations
#### Given Result:
The calculated t-statistic is [tex]\( t = 2.185 \)[/tex].
#### Step 3: Determine the Degrees of Freedom
The degrees of freedom (df) is calculated as:
[tex]\[ df = n - 1 \][/tex]
Given:
[tex]\[ df = 45 \][/tex]
#### Step 4: Determine p-values
The p-values for the one-tail and two-tail tests give information about the probability of observing the test statistic under the null hypothesis.
- One-tail [tex]\( p_{\text{val}} = 0.017 \)[/tex]
- Two-tail [tex]\( p_{\text{val}} = 0.034 \)[/tex]
#### Step 5: Critical Values
The critical values are determined from the t-distribution table for the given confidence level and degrees of freedom.
- Critical value (one-tail, 95%) = [tex]\( t_{\text{critical}} = 1.679 \)[/tex]
- Critical value (two-tail, 95%) = [tex]\( t_{\text{critical}} = 2.014 \)[/tex]
#### Step 6: Compare the Test Statistic with Critical Values
We compare our test statistic to these critical values to determine whether to reject the null hypothesis.
- Our test statistic [tex]\( t = 2.185 \)[/tex] exceeds the critical value for the two-tail test [tex]\( t_{\text{critical (two-tail)}} = 2.014 \)[/tex].
#### Step 7: Conclusion
Since [tex]\( t = 2.185 \)[/tex] is greater than [tex]\( t_{\text{critical (two-tail)}} = 2.014 \)[/tex], and the p-value for the two-tail test [tex]\( p_{\text{two-tail}} = 0.034 \)[/tex] is less than the significance level (typically 0.05 for a 95% confidence level), we reject the null hypothesis.
Conclusion: At a 95% confidence level, we have sufficient evidence to reject the null hypothesis and conclude that the mean miles driven differs significantly from 150 thousand miles.
### Hypothesis Testing for the Mean
We want to test if the mean miles driven (in thousands) by vehicles differs from a hypothesized mean.
Given Data:
1. Mean miles driven, [tex]\( \overline{x} = 150.7 \)[/tex]
2. Variance, [tex]\( s^2 = 4.661 \)[/tex]
3. Number of observations, [tex]\( n = 46 \)[/tex]
4. Hypothesized mean, [tex]\( \mu = 150 \)[/tex]
5. Degrees of freedom, [tex]\( df = 45 \)[/tex]
6. Test Statistic, [tex]\( t = 2.185 \)[/tex]
7. One-tail p-value, [tex]\( p_{\text{one-tail}} = 0.017 \)[/tex]
8. Two-tail p-value, [tex]\( p_{\text{two-tail}} = 0.034 \)[/tex]
9. Critical value (one-tail), [tex]\( t_{\text{critical (one-tail)}} = 1.679 \)[/tex]
10. Critical value (two-tail), [tex]\( t_{\text{critical (two-tail)}} = 2.014 \)[/tex]
11. Confidence Level, [tex]\( 95\% \)[/tex]
### Step-by-step Explanation:
#### Step 1: State the Hypotheses
We are performing a t-test to test the following hypotheses:
- Null hypothesis: [tex]\( H_0: \mu = 150 \)[/tex] (where [tex]\( \mu \)[/tex] is the population mean)
- Alternative hypothesis: [tex]\( H_a: \mu \neq 150 \)[/tex] (this is a two-tailed test)
#### Step 2: Calculate the Test Statistic
The test statistic is calculated using the formula for the t-test for a single sample:
[tex]\[ t = \frac{\overline{x} - \mu}{s / \sqrt{n}} \][/tex]
Where:
- [tex]\( \overline{x} \)[/tex] is the sample mean
- [tex]\( \mu \)[/tex] is the hypothesized population mean
- [tex]\( s \)[/tex] is the sample standard deviation (the square root of the variance)
- [tex]\( n \)[/tex] is the number of observations
#### Given Result:
The calculated t-statistic is [tex]\( t = 2.185 \)[/tex].
#### Step 3: Determine the Degrees of Freedom
The degrees of freedom (df) is calculated as:
[tex]\[ df = n - 1 \][/tex]
Given:
[tex]\[ df = 45 \][/tex]
#### Step 4: Determine p-values
The p-values for the one-tail and two-tail tests give information about the probability of observing the test statistic under the null hypothesis.
- One-tail [tex]\( p_{\text{val}} = 0.017 \)[/tex]
- Two-tail [tex]\( p_{\text{val}} = 0.034 \)[/tex]
#### Step 5: Critical Values
The critical values are determined from the t-distribution table for the given confidence level and degrees of freedom.
- Critical value (one-tail, 95%) = [tex]\( t_{\text{critical}} = 1.679 \)[/tex]
- Critical value (two-tail, 95%) = [tex]\( t_{\text{critical}} = 2.014 \)[/tex]
#### Step 6: Compare the Test Statistic with Critical Values
We compare our test statistic to these critical values to determine whether to reject the null hypothesis.
- Our test statistic [tex]\( t = 2.185 \)[/tex] exceeds the critical value for the two-tail test [tex]\( t_{\text{critical (two-tail)}} = 2.014 \)[/tex].
#### Step 7: Conclusion
Since [tex]\( t = 2.185 \)[/tex] is greater than [tex]\( t_{\text{critical (two-tail)}} = 2.014 \)[/tex], and the p-value for the two-tail test [tex]\( p_{\text{two-tail}} = 0.034 \)[/tex] is less than the significance level (typically 0.05 for a 95% confidence level), we reject the null hypothesis.
Conclusion: At a 95% confidence level, we have sufficient evidence to reject the null hypothesis and conclude that the mean miles driven differs significantly from 150 thousand miles.
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
