To test if there is a difference between the numbers of citizens in Durban and Johannesburg who would buy an electric vehicle, we can conduct a hypothesis test.
Here, the null hypothesis (H0) would be that the proportion of citizens in Durban who would buy an electric vehicle is the same as the proportion of citizens in Johannesburg. The alternative hypothesis (Ha) would be that the proportions are different.
H0: p1 = p2 (No difference between the proportions in Durban and Johannesburg)
Ha: p1 ≠ p2 (Difference between the proportions in Durban and Johannesburg)
To determine if there is a significant difference, we'll calculate the test statistic and compare it to the critical value at a significance level of 5%.
Given information:
- Sample size of Johannesburg (n1) = 200, proportion (p1) = 48%
- Sample size of Durban (n2) = 100, proportion (p2) = 52%
The formula to compute the test statistic (Z) is:
Z = (p1 - p2) / √((p(1-p)((1/n1) + (1/n2))))
Where p = (n1p1 + n2p2) / (n1 + n2)
Calculating p:
p = (200*0.48 + 100*0.52) / (200 + 100)
p = 88 / 300
p = 0.2933
Calculating the test statistic:
Z = (0.48 - 0.52) / √((0.2933(1-0.2933)((1/200) + (1/100))))
Z = -0.04 / √((0.2933(0.7067)((1/200) + (1/100))))
Z = -0.04 / √((0.2072)(0.01 + 0.02))
Z = -0.04 / √((0.2072)(0.03))
Z = -0.04 / √(0.06216)
Z = -0.04 / 0.2491
Z = -0.1607
The critical value for a two-tailed test at a significance level of 5% is approximately ±1.96.
Since the test statistic (-0.1607) is between -1.96 and +1.96, we fail to reject the null hypothesis. There is not enough evidence to suggest a significant difference between the numbers of citizens in Durban and Johannesburg who would buy an electric vehicle.
A small opinion poll among 200 Johannesburg and 100 Durban citizens indicated that 48% and 52% respectively, would buy an electrical vehicle if the price is less than R16 000. Is there a difference between the numbers of citizens in Durban that would buy an electrical vehicle than in Johannesburg? Test at a significance level of 5% if the populations are independent and normally distributed. Give the critical value.
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