Question 1184272: A health officer is trying to study the malaria situation of Zambia. From the records of
seasonal blood survey (SBS) results he came to understand that the proportion of people
having malaria in Zambia was 3.8% in 2015. The size of the sample considered was 15,000.
He also realized that during the year that followed (2016), blood samples were taken from
10,000 randomly selected persons. The result of the 2016 seasonal blood survey showed
that 200 persons were positive for malaria. Help the Health officer in testing the hypothesis
that the malaria situation of 2016 did not show any significant difference from that of 2015
(take the 5% level of significance).
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to perform a hypothesis test to determine if the malaria situation in 2016 was significantly different from 2015:
**1. State the Hypotheses:**
* **Null Hypothesis (H0):** There is no significant difference in the proportion of people with malaria between 2015 and 2016. (p1 = p2)
* **Alternative Hypothesis (H1):** There is a significant difference in the proportion of people with malaria between 2015 and 2016. (p1 ≠ p2)
**2. Define the Significance Level:**
α = 0.05 (5%)
**3. Calculate the Sample Proportions:**
* **2015:**
- p1 = 0.038 (3.8%)
- n1 = 15,000
* **2016:**
- Number of positive cases = 200
- p2 = 200 / 10,000 = 0.02 (2%)
- n2 = 10,000
**4. Calculate the Test Statistic (z-score):**
We'll use a two-proportion z-test. The formula for the test statistic is:
z = (p1 - p2) / sqrt[ p(1-p) * (1/n1 + 1/n2) ]
Where 'p' is the pooled proportion, calculated as:
p = (x1 + x2) / (n1 + n2) = (0.038 * 15000 + 200) / (15000 + 10000) = (570 + 200) / 25000 = 770 / 25000 = 0.0308
Now, plug the values into the z-score formula:
z = (0.038 - 0.02) / sqrt[ 0.0308 * (1 - 0.0308) * (1/15000 + 1/10000) ]
z = 0.018 / sqrt[ 0.0298 * (0.0000667 + 0.0001) ]
z = 0.018 / sqrt(0.0298 * 0.0001667)
z = 0.018 / sqrt(0.000004967)
z = 0.018 / 0.002229
z ≈ 8.08
**5. Determine the Critical Value:**
Since this is a two-tailed test (H1: p1 ≠ p2) at α = 0.05, we need to find the critical z-values that correspond to the tails of the standard normal distribution. Look up z = ±1.96.
**6. Make a Decision:**
* **Compare the test statistic to the critical values:** Our calculated z-score (8.08) is much larger than the critical value (1.96).
* **Conclusion:** Because the absolute value of the calculated z-score is greater than the critical value, we reject the null hypothesis.
**7. Interpret the Results:**
There is sufficient evidence at the 5% level of significance to conclude that there is a statistically significant difference in the proportion of people with malaria between 2015 and 2016 in Zambia. The malaria situation in 2016 was significantly different from 2015.
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