A sample of 400 items is taken from a population whose standard deviation is
10. The mean of the sample is 40. Compute whether the sample has come from a
population with mean 38. Also calculate 95% confidence interval for mean of the
population
To determine if the sample has come from a population with mean 38, we can use a hypothesis test. The hypothesis test involves comparing the sample mean with the population mean and considering the standard deviation.
First, let's state the null hypothesis (H0) and the alternative hypothesis (Ha):
H0: The sample mean is equal to the population mean (μ = 38)
Ha: The sample mean is not equal to the population mean (μ ≠ 38)
Next, we can calculate the test statistic, which is the z-score:
z = (x̄ - μ) / (σ / √n)
Where:
x̄ = sample mean
μ = population mean
σ = population standard deviation
n = sample size
Given:
x̄ = 40
μ = 38
σ = 10
n = 400
Plugging in the values, we can calculate the test statistic:
z = (40 - 38) / (10 / √400)
z = 2 / (10 / 20)
z = 2 / 0.5
z = 4
The next step is to determine the critical value for a 95% confidence level. For a two-tailed test, we divide the significance level (α) by 2, giving us α/2 = 0.025. Looking up the z-value for a 95% confidence level, we find it to be approximately 1.96.
Since the calculated test statistic (4) is greater than the critical value (1.96), we reject the null hypothesis.
Therefore, we can conclude that the sample did not come from a population with a mean of 38.
Now let's calculate the 95% confidence interval for the mean of the population.
The formula for the confidence interval is:
CI = x̄ ± (z * (σ / √n))
Plugging in the values:
CI = 40 ± (1.96 * (10 / √400))
Simplifying:
CI = 40 ± (1.96 * (10 / 20))
CI = 40 ± (1.96 * 0.5)
CI = 40 ± 0.98
The 95% confidence interval for the mean of the population is (39.02, 40.98).
This means that we are 95% confident that the true population mean falls within this range.
A sample of 400 items is taken from a population whose standard deviation is
10. The mean of the sample is 40. Compute whether the sample has come from a
population with mean 38. Also calculate 95% confidence interval for mean of the
population