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.
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.
To test if there is a difference between the numbers of citizens in Durban and Johannesburg who would buy an electric vehicle, we can perform a hypothesis test.
Hypotheses:
Null hypothesis (H0): The proportion of citizens in Durban who would buy an electric vehicle is equal to the proportion of citizens in Johannesburg who would buy an electric vehicle.
Alternative hypothesis (Ha): The proportion of citizens in Durban who would buy an electric vehicle is not equal to the proportion of citizens in Johannesburg who would buy an electric vehicle.
To perform the test, we will calculate the test statistic and compare it to the critical value. The test statistic follows a normal distribution when samples are large and independent.
Here are the calculations we need:
Step 1: Calculate the standard error for each sample proportion.
For Johannesburg:
Standard Error(Johannesburg) = sqrt((0.48 * (1-0.48)) / 200)
For Durban:
Standard Error(Durban) = sqrt((0.52 * (1-0.52)) / 100)
Step 2: Calculate the test statistic (Z-score) for the difference in sample proportions.
Z = (proportion(Johannesburg) - proportion(Durban)) / sqrt(Standard Error(Johannesburg)^2 + Standard Error(Durban)^2)
Step 3: Compare the absolute value of the calculated Z-score with the critical value.
To find the critical value at a significance level of 5%, we need to use a Z-table or calculator. The critical value depends on whether the test is one-tailed or two-tailed.
Since the alternative hypothesis is two-tailed, we will divide the significance level (5%) by 2 to get a critical value of 2.5%.
Looking up the critical value in the Z-table, z_α/2, we find it to be 1.96.
Therefore, the critical value for this hypothesis test is 1.96.
Now, you can use the calculated test statistic (Z-score) to determine if there is a significant difference between the proportions of citizens in Durban and Johannesburg who would buy an electric vehicle.
To test if there is a difference between the numbers of citizens in Durban and Johannesburg who would buy an electric vehicle, we can use a hypothesis testing approach. The null hypothesis (H0) is that there is no difference between the two populations, and the alternative hypothesis (HA) is that there is a difference.
Let's denote the proportions of citizens in Johannesburg and Durban who would buy an electric vehicle as p1 and p2, respectively. The sample sizes are n1 = 200 for Johannesburg and n2 = 100 for Durban.
To test the hypothesis, we will use the z-test for two independent proportions. The test statistic is given by:
z = (p1 - p2) / √(p̂(1-p̂)(1/n1 + 1/n2))
where p̂ is the pooled proportion, given by:
p̂ = (x1 + x2) / (n1 + n2)
and x1 and x2 are the numbers of citizens in Johannesburg and Durban, respectively, who would buy an electric vehicle.
Given that 48% of citizens in Johannesburg would buy an electric vehicle, we have x1 = 0.48 * 200 = 96. Similarly, for Durban, 52% would buy an electric vehicle, so x2 = 0.52 * 100 = 52.
Using these values, we can calculate p̂ and then the z-value.
p̂ = (96 + 52) / (200 + 100) = 0.5
z = (0.48 - 0.52) / √(0.5*(1-0.5)(1/200 + 1/100))
Now, to test the hypothesis at a significance level of 5% (or 0.05), we need to determine the critical value. The critical value corresponds to the z-score that defines the rejection region for the null hypothesis.
For a two-tailed test (since we are testing for a difference, not a specific direction), a significance level of 5% will be split equally among the two tails. So, we need to find the z-score that corresponds to a cumulative probability of 0.025 in each tail. This can be done using a standard normal distribution table or a statistical calculator.
By using the standard normal distribution table or a calculator, we find that the critical value for a two-tailed test at a significance level of 5% (or 0.05) is approximately 1.96.
Therefore, the critical value for this hypothesis test is 1.96. To complete the test, we would compare the calculated z-value with this critical value to make a decision about whether to reject or fail to reject the null hypothesis.