Question 1185822: A reading coordinator in a large public school system suspect that poor readers may test lower in IQ than children whose reading is satisfactory. He draws a random sample of 30, fifth grade students who are poor readers. Historically fifth grade students in the students in the school system have had an average IQ of 105. The sample of 30 has XBAR =101.5 and s(XBAR)=1.42. Test the appropriate hypothesis at the 5% level.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to conduct a hypothesis test to determine if poor readers have lower IQ scores:
**1. Hypotheses:**
* **Null Hypothesis (H0):** Poor readers have the same average IQ as other students (μ = 105).
* **Alternative Hypothesis (H1):** Poor readers have a lower average IQ than other students (μ < 105). This is a left-tailed test.
**2. Significance Level (alpha):** α = 0.05 (5%)
**3. Test Statistic:** Since the sample size is small (n = 30) and the population standard deviation is unknown, we use a t-test. The test statistic is:
```
t = (x̄ - μ) / (s / √n)
```
Where:
* x̄ is the sample mean (101.5)
* μ is the population mean under H0 (105)
* s is the sample standard deviation (1.42)
* n is the sample size (30)
**4. Degrees of Freedom:** df = n - 1 = 30 - 1 = 29
**5. Critical Value:** For a one-tailed (left-tailed) t-test with α = 0.05 and df = 29, we consult a t-table or calculator. The critical value is approximately t = -1.699.
**6. Decision Rule:** Reject H0 if the calculated t-statistic is less than the critical value (-1.699).
**7. Calculation:**
```
t = (101.5 - 105) / (1.42 / √30)
t = -3.5 / 0.259
t ≈ -13.51
```
**8. Conclusion:**
The calculated t-statistic (-13.51) is *less than* the critical value (-1.699). Therefore, we *reject* the null hypothesis.
**Interpretation:**
At a 5% significance level, there is sufficient evidence from the sample to conclude that poor readers have a lower average IQ than other students in the school system.
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