In a random sample of 1000 persons from town A, 400 are found to be consumers of wheat. In a random of 800 from town B, 400 are found to be consumers of wheat, do these data reveal a significant difference between town A and town B, so far as the proportion of wheat consumers is concerned?
To determine if there is a significant difference between town A and town B in terms of the proportion of wheat consumers, we can conduct a hypothesis test for the difference in proportions.
Let p1 be the proportion of wheat consumers in town A and p2 be the proportion of wheat consumers in town B.
Null hypothesis (H0): p1 = p2
Alternative hypothesis (H1): p1 ≠ p2
Let's calculate the proportions of wheat consumers in each town:
p1 = 400/1000 = 0.4
p2 = 400/800 = 0.5
Next, we calculate the standard error of the difference in proportions:
SE = sqrt[(p1(1-p1)/n1) + (p2(1-p2)/n2)]
SE = sqrt[(0.4(1-0.4)/1000) + (0.5(1-0.5)/800)]
SE = sqrt[(0.24/1000) + (0.25/800)]
SE = sqrt[0.00024 + 0.0003125]
SE = sqrt(0.0005525)
SE ≈ 0.0235
Now, we calculate the z-score for the difference in proportions:
z = (p1 - p2) / SE
z = (0.4 - 0.5) / 0.0235
z = -0.1 / 0.0235
z ≈ -4.26
Using a standard normal distribution table, we find that the z-value of -4.26 corresponds to a very small p-value (much smaller than 0.01). This means that we reject the null hypothesis and conclude that there is a significant difference between town A and town B in terms of the proportion of wheat consumers.
Therefore, based on the data provided, there is a significant difference between town A and town B in terms of the proportion of wheat consumers.