Math Problem Statement
A manufacturing company wants to compare the average production output of two of its plants. Plant X has a sample of 40 days with an average daily production of 150 units and a standard deviation of 15 units. Plant Y has a sample of 35 days with an average daily production of 145 units and a standard deviation of 12 units. Test at the 1% significance level whether there is a significant difference in the average daily production output between the two plants.
Solution
To test whether there is a significant difference in the average daily production output between the two plants, we can conduct a two-sample t-test for the difference in means. Given that the sample sizes are relatively large (40 and 35), and we assume the populations are normally distributed, a two-sample t-test is appropriate.
Step-by-Step Solution:
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Hypotheses:
- Null hypothesis (): There is no significant difference in the average production output between the two plants.
- Alternative hypothesis (): There is a significant difference in the average production output between the two plants. This is a two-tailed test because we are testing whether the means are different, regardless of direction.
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Significance Level:
- (1% significance level)
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Test Statistic (t-score):
The formula for the t-score for two independent samples is: where:
- = sample mean of Plant X = 150
- = sample mean of Plant Y = 145
- = sample standard deviation of Plant X = 15
- = sample standard deviation of Plant Y = 12
- = sample size of Plant X = 40
- = sample size of Plant Y = 35
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Degrees of Freedom (df):
The degrees of freedom for the two-sample t-test are calculated using the following formula (Welch-Satterthwaite equation for unequal variances):
Now, let's compute the values.
I'll calculate the test statistic and degrees of freedom .The calculated t-statistic is approximately , and the degrees of freedom (df) are approximately 72.45.
- Critical Value and p-value:
Since this is a two-tailed test at the 1% significance level (), we will look for the critical t-value corresponding to in a t-distribution with approximately 72 degrees of freedom.
Now, I will find the critical t-value and compare it to the calculated t-statistic to draw a conclusion.The critical t-value at the 1% significance level (two-tailed) is approximately . The p-value corresponding to the calculated t-statistic is approximately .
Conclusion:
- The calculated t-statistic () is less than the critical t-value ().
- The p-value () is greater than the significance level ().
Since the t-statistic does not exceed the critical value and the p-value is greater than 0.01, we fail to reject the null hypothesis. There is no significant difference in the average daily production output between Plant X and Plant Y at the 1% significance level.
Would you like more details or have any questions? Here are some related questions to explore:
- What assumptions are required for a two-sample t-test?
- How would the results change if we used a 5% significance level?
- What is the interpretation of the p-value in hypothesis testing?
- How do we handle cases where the sample sizes or variances are significantly different?
- What are the implications of failing to reject the null hypothesis in a business setting?
Tip: When conducting hypothesis tests, always ensure the assumptions of the test (e.g., normality, independence) are met for valid results.
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Math Problem Analysis
Mathematical Concepts
Statistics
Hypothesis Testing
Two-sample t-test
Formulas
t = (X̄ - Ȳ) / √[(s_X^2 / n_X) + (s_Y^2 / n_Y)]
Degrees of freedom (df) = [(s_X^2 / n_X + s_Y^2 / n_Y)^2] / [(s_X^2 / n_X)^2 / (n_X - 1) + (s_Y^2 / n_Y)^2 / (n_Y - 1)]
Theorems
Welch's t-test
Central Limit Theorem
Suitable Grade Level
Undergraduate Statistics
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