Question 1193376: The quality manager of a washing machine company feels that the mean time between failures of the motors received is at most 90 days. The quality manager wants to test his intuition. Hence , he has taken a sample of 25 motors whose mean time between failure and its variance are found to be 93 days and 16 days , respectively. Verify the intuition of the quality manager at a significance level of 0.05.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! The problem is about hypothesis testing, specifically a one-sample t-test. Here's how we can approach it:
1. Set up the null and alternative hypotheses.
2. Calculate the t-statistic.
3. Determine the critical t-value.
4. Compare the calculated and critical t-value to make a decision.
5. Interpret the results to verify the quality manager's intuition.
Let's proceed with the analysis.
```python?code_reference&code_event_index=3
import scipy.stats as stats
import math
# Define variables
sample_mean = 93
sample_variance = 16
sample_size = 25
hypothesized_mean = 90
significance_level = 0.05
# Calculate the t-statistic
t_statistic = (sample_mean - hypothesized_mean) / (math.sqrt(sample_variance) / math.sqrt(sample_size))
# Calculate degrees of freedom
degrees_of_freedom = sample_size - 1
# Calculate the critical t-value
critical_t_value = stats.t.ppf(1 - significance_level, degrees_of_freedom)
# Compare t-statistic with critical t-value
if t_statistic > critical_t_value:
print("Reject the null hypothesis")
else:
print("Fail to reject the null hypothesis")
# Print the results
print("T-statistic:", t_statistic)
print("Critical T-value:", critical_t_value)
```
```text?code_stdout&code_event_index=3
Reject the null hypothesis
T-statistic: 3.75
Critical T-value: 1.7108820799094275
```
**1. Set up Hypotheses**
* **Null Hypothesis (H0):** The mean time between failures of the motors is at most 90 days.
* μ ≤ 90
* **Alternative Hypothesis (H1):** The mean time between failures of the motors is greater than 90 days.
* μ > 90
**2. Determine the Test Statistic**
Since we have one sample and we do not know the population variance, we will use the **one-sample t-test**.
**3. Calculate the Test Statistic**
* **Sample Data:**
* Sample size (n) = 25
* Sample mean (x̄) = 93 days
* Sample variance (s^2) = 16 days
**4. Calculate the t-statistic:**
t = (x̄ - μ) / (s / √n)
t = (93 - 90) / (√16 / √25)
t = 3 / (4 / 5)
t = 3.75
**5. Determine the Critical Value**
* Significance level (α) = 0.05
* This is a one-tailed test (since H1 is μ > 90)
* Degrees of freedom (df) = n - 1 = 25 - 1 = 24
* Find the critical t-value from the t-distribution table or using a statistical software:
* tα,df = t0.05,24 ≈ 1.711
**6. Decision Rule**
* Reject H0 if the calculated t-statistic is greater than the critical t-value.
* Otherwise, fail to reject H0.
**7. Conclusion**
* Calculated t-statistic (3.75) is greater than the critical t-value (1.711).
* Therefore, **we reject the null hypothesis (H0).**
**Interpretation:**
There is sufficient evidence at the 0.05 significance level to conclude that the mean time between failures of the motors is greater than 90 days. This contradicts the quality manager's intuition that the mean time between failures is at most 90 days.
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