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Question 1193539: (1) The weight of cement bags produced in a cement company follow normal distribution
whose population is infinite. The expected mean of weight of the cement bags for sales of
this population is 45KG and its variance is unknown. The sales manager of the firm
claims that the mean weight of the cement bags is significantly more than the expected
mean weight of the population. So, he has selected a random sample of 36 bags and its
mean and variance are found to be 44KG and 1.25KG, respectively. Verify the intuition
of the sales manager at a significance level of 0.05.
(2) 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 ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! Certainly, let's analyze both scenarios.
**1. Cement Bag Weights**
**a) Set up Hypotheses**
* **Null Hypothesis (H0):** The mean weight of the cement bags is less than or equal to the expected mean weight.
* μ ≤ 45 kg
* **Alternative Hypothesis (H1):** The mean weight of the cement bags is significantly greater than the expected mean weight.
* μ > 45 kg
**b) Choose the Test Statistic**
* Since the population variance is unknown and the sample size is greater than 30, we can use the **one-sample t-test**.
**c) Calculate the Test Statistic**
* **Given:**
* Sample size (n): 36
* Sample mean (x̄): 44 kg
* Sample variance (s²): 1.25 kg²
* Population mean (μ₀): 45 kg
* **Calculate the standard error:**
* Standard Error (SE) = s / √n
* SE = √(1.25) / √36
* SE ≈ 0.2041
* **Calculate the t-score:**
* t = (x̄ - μ₀) / SE
* t = (44 - 45) / 0.2041
* t ≈ -4.898
**d) Determine Critical Value**
* **Significance Level:** α = 0.05
* **Degrees of Freedom (df):** n - 1 = 36 - 1 = 35
* **One-tailed test (right-tailed):** Find the critical t-value from a t-distribution table.
* t_critical ≈ 1.690
**e) Decision Rule**
* If the calculated t-score is greater than the critical t-value, reject the null hypothesis.
* If the calculated t-score is less than or equal to the critical t-value, fail to reject the null hypothesis.
**f) Make a Decision**
* Our calculated t-score (-4.898) is less than the critical t-value (1.690).
* **Conclusion:** We fail to reject the null hypothesis.
**Interpretation**
The evidence does not support the sales manager's claim that the mean weight of the cement bags is significantly greater than the expected mean weight.
**2. Washing Machine Motor Time Between Failures**
**a) Set up Hypotheses**
* **Null Hypothesis (H0):** The mean time between failures is less than or equal to 90 days.
* μ ≤ 90 days
* **Alternative Hypothesis (H1):** The mean time between failures is greater than 90 days.
* μ > 90 days
**b) Choose the Test Statistic**
* Similar to the first scenario, we can use the **one-sample t-test** since the population variance is unknown.
**c) Calculate the Test Statistic**
* **Given:**
* Sample size (n): 25
* Sample mean (x̄): 93 days
* Sample variance (s²): 16 days²
* Population mean (μ₀): 90 days
* **Calculate the standard error:**
* SE = s / √n
* SE = √(16) / √25
* SE = 4 / 5
* SE = 0.8 days
* **Calculate the t-score:**
* t = (x̄ - μ₀) / SE
* t = (93 - 90) / 0.8
* t = 3 / 0.8
* t = 3.75
**d) Determine Critical Value**
* **Significance Level:** α = 0.05
* **Degrees of Freedom (df):** n - 1 = 25 - 1 = 24
* **One-tailed test (right-tailed):** Find the critical t-value from a t-distribution table.
* t_critical ≈ 1.711
**e) Decision Rule**
* If the calculated t-score is greater than the critical t-value, reject the null hypothesis.
* If the calculated t-score is less than or equal to the critical t-value, fail to reject the null hypothesis.
**f) Make a Decision**
* Our calculated t-score (3.75) is greater than the critical t-value (1.711).
* **Conclusion:** We reject the null hypothesis.
**Interpretation**
The evidence supports the quality manager's intuition that the mean time between failures of the motors is significantly greater than 90 days.
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