(b) A sample of 900 members has a mean of 3.4cm and standard deviation of 2.61cm. Is the
sample from a large population of mean 3.25cm and standard deviation 2.26cm?
To determine if the sample is from a large population, we can use hypothesis testing. We can set up the following hypotheses:
Null Hypothesis (H0): The sample is from a large population of mean 3.25cm and standard deviation 2.26cm.
Alternative Hypothesis (Ha): The sample is not from a large population of mean 3.25cm and standard deviation 2.26cm.
We can perform a Z-test to test these hypotheses. The formula for the Z-test statistic is:
Z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
In this case:
sample mean = 3.4cm
population mean = 3.25cm
population standard deviation = 2.26cm
sample size = 900
Calculating the Z-test statistic:
Z = (3.4 - 3.25) / (2.26 / sqrt(900))
Z = (0.15) / (2.26 / 30)
Z = 0.15 / 0.0773
Z ≈ 1.938
Now, we need to determine the p-value associated with this Z-test statistic. To do this, we can use a Z-table or calculator.
Assuming a significance level of 0.05 (or 5%), we compare the p-value to the significance level to make our decision.
If the p-value is less than or equal to 0.05, we reject the null hypothesis and conclude that the sample is not from a large population. If the p-value is greater than 0.05, we fail to reject the null hypothesis and do not have enough evidence to conclude that the sample is not from a large population.
Using a Z-table or calculator, we find that the p-value associated with a Z-test statistic of 1.938 is approximately 0.053.
Since the p-value (0.053) is greater than 0.05, we fail to reject the null hypothesis. Therefore, we do not have enough evidence to conclude that the sample is not from a large population.