A sample of 400 items is taken from a population whose standard deviation is
10. The mean of the sample is 40. Test whether the sample has come from a
population with mean 38. Also calculate 95% confidence interval for the
population.
To test whether the sample has come from a population with a mean of 38, we can use a hypothesis test. The null hypothesis, denoted as H0, assumes that the population mean is 38, while the alternative hypothesis, denoted as Ha, assumes that the population mean is not 38.
Step 1: Define the null and alternative hypotheses:
H0: μ = 38 (Population mean is 38)
Ha: μ ≠ 38 (Population mean is not 38)
Step 2: Choose the significance level (α):
We are given the confidence level of 95%. Since α = 1 - Confidence Level, in this case, α = 1 - 0.95 = 0.05.
Step 3: Calculate the test statistic:
The test statistic for a hypothesis test comparing a sample mean to a population mean is calculated as:
𝑧 = (𝑥̅ - 𝑢) / (𝜎 / √𝑛)
Where 𝑥̅ is the sample mean, 𝑢 is the population mean, 𝜎 is the population standard deviation, and 𝑛 is the sample size.
In this case, 𝑥̅ = 40, 𝑢 = 38, 𝜎 = 10, and 𝑛 = 400:
𝑧 = (40 - 38) / (10 / √400) = 2 / 0.5 = 4
Step 4: Determine the critical value:
Since we are doing a two-tailed test, we need to look up the critical value that corresponds to the significance level (α / 2 = 0.05 / 2 = 0.025) in the standard normal distribution.
Looking up the value in a standard normal distribution table, we find that the critical value is approximately ±1.96.
Step 5: Make a decision:
If the test statistic is outside the range defined by the critical values, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
In our case, the test statistic is 4, which is larger than 1.96. Therefore, we reject the null hypothesis.
Conclusion: There is sufficient evidence to conclude that the sample did not come from a population with a mean of 38.
To calculate the 95% confidence interval for the population mean, we can use the formula:
Confidence interval = 𝑥̅ ± (𝑧 * (𝜎 / √𝑛))
Using the values we have, 𝑥̅ = 40, 𝑧 = 1.96 (corresponding to a 95% confidence level), 𝜎 = 10, and 𝑛 = 400:
Confidence interval = 40 ± (1.96 * (10 / √400)) = 40 ± (1.96 * (10 / 20)) = 40 ± 0.98
Therefore, the 95% confidence interval for the population mean is (39.02, 40.98).