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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.
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