Question 1193375: 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 bag 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 significant level of 0.05.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **1. Set up Hypotheses**
* **Null Hypothesis (H0):** The mean weight of the cement bags is equal to the expected population mean.
* μ = 45 kg
* **Alternative Hypothesis (H1):** The mean weight of the cement bags is significantly greater than the expected population mean.
* μ > 45 kg
**2. Determine Test Statistic**
* Since the population variance is unknown and the sample size is large (n = 36), we will use a **t-test**.
* **Calculate the t-statistic:**
* t = (sample mean - population mean) / (sample standard deviation / √sample size)
* t = (44 - 45) / (√1.25 / √36)
* t = -1 / (1.118 / 6)
* t = -5.37
**3. Determine Degrees of Freedom**
* Degrees of freedom (df) = sample size - 1 = 36 - 1 = 35
**4. Find the Critical Value**
* Using a t-distribution table or a statistical software (like Python's scipy library), find the critical value for a one-tailed t-test with 35 degrees of freedom and a significance level of 0.05.
* The critical value for a one-tailed t-test at 0.05 significance level with 35 degrees of freedom is approximately 1.690.
**5. Compare Test Statistic to Critical Value**
* Calculated t-statistic (-5.37) < Critical value (1.690)
**6. Make a Decision**
* Since the calculated t-statistic is less than the critical value, we **fail to reject the null hypothesis**.
**7. Conclusion**
* There is **not enough evidence** to support the sales manager's claim that the mean weight of the cement bags is significantly greater than the expected population mean at a 0.05 significance level.
**In Summary:**
The statistical analysis does not support the sales manager's claim. The sample mean of 44 kg is lower than the expected population mean of 45 kg, and the t-test does not provide sufficient evidence to conclude that the difference is statistically significant.
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