It is believed that at least 60% of voters from a certain region in Canada favor the free trade agreement (FTA). A recent poll indicated that out of 400 randomly selected individuals, 250 favored the FTA. If we wished to perform a test to determine whether the proportion of those favoring the FTA is greater than 60%, at the 5% level of significance, we would:
Fail to reject H0 since the calculated value of the test statistic is 1.033 which is less than 1.645.
Fail to reject H0 since the calculated value of the test statistic is 1.033 which is less than 1.96.
Fail to reject H0 since the calculated value of the test statistic is 1.0204 which is less than 1.96.
Fail to reject H0 since the calculated value of the test statistic is 1.0204 which is less than 1.645.
Not need to test since everyone knows that FTA is good.
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A seed company claims that 80% of the seeds of a certain variety of tomato will germinate if sown under normal growing conditions. A government inspector is interested in whether or not the proportion of seeds germinating is living up to the company's claim. He randomly selects a sample of 200 seeds from a large shipment and tests the sample for percentage germination. If 155 of the 200 seeds germinate, then the calculated value of the test statistic used to test the hypothesis of interest is:
- .847
-.884
-.897
-.825
-.858
Fail to reject H0 since the calculated value of the test statistic is 1.0204 which is less than 1.645.
Using a formula for a binomial proportion one-sample z-test with your data included, we have:
z = .625 - .60 -->test value (250/400 = .625) minus population value (.60)
divided by
√[(.60)(.40)/400]
Calculating, z = 1.02
Critical value = 1.645 (one-tailed)
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Test statistic for your second problem can be calculated the same way:
z = .775 - .80 -->test value (155/200 = .775) minus population value (.80)
divided by
√[(.80)(.20)/200]
Calculating, z = -0.884
To determine the calculated value of the test statistic, we need to calculate the standard error and the observed proportion.
The standard error can be calculated using the formula:
Standard Error = sqrt((p * (1 - p)) / n)
Where p is the proportion of seeds germinating according to the company's claim (80%) and n is the sample size (200).
Plugging in the values, we get:
Standard Error = sqrt((0.8 * (1 - 0.8)) / 200)
= sqrt(0.16 / 200)
= sqrt(0.0008)
≈ 0.0283
Next, we calculate the observed proportion, which is the proportion of seeds that germinated in the sample (155/200):
Observed Proportion = 155 / 200
= 0.775
The test statistic can be calculated using the formula:
Test Statistic = (observed proportion - hypothesized proportion) / standard error
Where the hypothesized proportion is the company's claim (80%).
Plugging in the values, we get:
Test Statistic = (0.775 - 0.8) / 0.0283
= -0.884
Therefore, the calculated value of the test statistic is -0.884.
To solve the first question, we need to conduct a hypothesis test. The null hypothesis (H0) is that the proportion of voters in favor of the free trade agreement is 60% or less. The alternative hypothesis (Ha) is that the proportion is greater than 60%.
To calculate the test statistic, we use the formula:
test statistic = (sample proportion - null hypothesis proportion) / standard error
The sample proportion is 250/400 = 0.625
The null hypothesis proportion is 0.60
To calculate the standard error, we use the formula:
standard error = sqrt((null hypothesis proportion * (1 - null hypothesis proportion)) / sample size)
The sample size is 400.
Plugging the values into the formula, we get:
standard error = sqrt((0.60 * (1 - 0.60)) / 400) = 0.0204
Finally, we calculate the test statistic:
test statistic = (0.625 - 0.60) / 0.0204 = 1.2255
To determine whether to reject or fail to reject the null hypothesis, we compare the test statistic to the critical value at the 5% level of significance. Since we are testing whether the proportion is greater, we use the one-tail critical value.
At the 5% level of significance, the critical value is 1.645.
Comparing the test statistic to the critical value, we find that 1.2255 is less than 1.645.
Therefore, the correct answer is: Fail to reject H0 since the calculated value of the test statistic is 1.2255 which is less than 1.645.
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To solve the second question, we need to calculate the test statistic for testing whether the proportion of seeds germinating is living up to the company's claim.
Again, we use the formula:
test statistic = (sample proportion - null hypothesis proportion) / standard error
The sample proportion is 155/200 = 0.775
The null hypothesis proportion is 0.80
To calculate the standard error, we use the same formula as before:
standard error = sqrt((null hypothesis proportion * (1 - null hypothesis proportion)) / sample size)
The sample size is 200.
Plugging the values into the formula, we get:
standard error = sqrt((0.80 * (1 - 0.80)) / 200) = 0.025
Finally, we calculate the test statistic:
test statistic = (0.775 - 0.80) / 0.025 = -1.00
Therefore, the calculated value of the test statistic is -1.00.
The correct answer is: -1.00