SOLUTION: A lab technician is tested for her consistency by taking multiple measurements of cholesterol levels from the same blood sample. The target accuracy is a standard deviation less th

Question 1193999: A lab technician is tested for her consistency by taking multiple measurements of cholesterol levels from the same blood sample. The target accuracy is a standard deviation less than 1.2. If 20 measurements are taken and their standard deviation is 1.1, is there enough evidence at a=.05 that she is meeting the target accuracy?

The correct Null and Alternative Hypothesis statement would be?
The correct critical value would be?
The correct test value would be?
The decision and summary statement would be?

Answer by parmen(42) (Show Source): You can put this solution on YOUR website!
**1. Set up Hypotheses**
* **Null Hypothesis (H₀):** σ² ≥ 1.2² (The population variance of the technician's measurements is greater than or equal to 1.44)
* **Alternative Hypothesis (H₁):** σ² < 1.2² (The population variance of the technician's measurements is less than 1.44)
**2. Determine the Test Statistic**
* We will use the Chi-Square test statistic for testing the population variance.
* **Test Statistic:** χ² = (n - 1) * s² / σ₀²
* where:
* n = sample size (20 measurements)
* s² = sample variance (1.1² = 1.21)
* σ₀² = hypothesized population variance (1.2² = 1.44)
* **Calculate the Test Statistic:**
* χ² = (20 - 1) * 1.21 / 1.44 = 19 * 1.21 / 1.44 = 15.96
**3. Determine the Critical Value**
* **Degrees of Freedom (df):** df = n - 1 = 20 - 1 = 19
* **Significance Level (α):** α = 0.05
* **Critical Value:**
* Since this is a left-tailed test, we find the critical value from the Chi-Square distribution table with 19 degrees of freedom and an area of 0.05 in the left tail.
* Using a Chi-Square table or statistical software, the critical value for χ²(0.05, 19) is approximately 8.907.
**4. Decision**
* **Compare Test Statistic to Critical Value:**
* Our calculated test statistic (15.96) is greater than the critical value (8.907).
* **Decision:**
* Since the test statistic does not fall in the rejection region (less than the critical value), we **fail to reject the null hypothesis**.
**5. Summary Statement**
* There is **not enough evidence** at the 0.05 significance level to conclude that the lab technician is meeting the target accuracy of a standard deviation less than 1.2.
**Key Points:**
* We use the Chi-Square test for testing hypotheses about population variance.
* The critical value is determined based on the significance level and degrees of freedom.
* We compare the test statistic to the critical value to make a decision about the null hypothesis.

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