A study was done on skull sizes of humans during different time periods. The results are shown in the table. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) and (b) below.
Use a 0.050.05 significance level, and test the claim that the mean skull breadth in 4000 B.C is less than the mean skull breadth in A.D 150.
4000 B.C
u1
n=30
x bar =130.94mm
s=5.49 mn
AD 150
u2
n=30
x bar= 135.23
s= 5.41mn
null is u1=u2
ater is u1<u2
t= -3.05
p-value .002
I am having a problem figuring out the confidence interval. I do not what i am doing wrong.
To calculate the confidence interval, we need the sample mean, standard deviation, sample size, and the appropriate critical value based on the significance level. In this case, the confidence level is not provided, so we will assume a commonly used confidence level of 95%, which corresponds to a significance level of 0.05 (two-tailed test).
Now, to calculate the confidence interval, we can follow these steps:
Step 1: Determine the critical value.
Since we assume a 95% confidence level, we need to find the critical value for a two-tailed test at a significance level of 0.05. The critical value can be obtained from a t-distribution table or calculated using software. In this case, with a sample size of 30, the critical value is approximately 2.045.
Step 2: Calculate the standard error.
The standard error can be calculated using the formula:
standard error = sqrt((s1^2 / n1) + (s2^2 / n2)), where s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
In this case, s1 = 5.49, n1 = 30, s2 = 5.41, and n2 = 30.
Substituting these values into the formula, we get:
standard error = sqrt((5.49^2 / 30) + (5.41^2 / 30)) ≈ 0.886
Step 3: Calculate the margin of error.
The margin of error is the product of the critical value and the standard error.
In this case, the margin of error = 2.045 * 0.886 ≈ 1.81
Step 4: Calculate the confidence interval.
The confidence interval is calculated by subtracting the margin of error from the sample mean and adding it to the sample mean.
In this case, the confidence interval is given by:
(130.94 - 1.81, 130.94 + 1.81)
which simplifies to:
(129.13, 132.75)
Therefore, the 95% confidence interval for the mean skull breadth of humans in 4000 B.C. is approximately (129.13mm, 132.75mm).
To calculate the confidence interval, you can follow these steps:
Step 1: Determine the level of confidence. Let's assume a 95% confidence level.
Step 2: Identify the standard deviations and sample sizes for each group. In this case, the standard deviation for 4000 B.C is 5.49 mm (s1) and for AD 150 is 5.41 mm (s2), with sample sizes of 30 (n1) for both groups.
Step 3: Calculate the standard error (SE) for the difference between the means using the formula:
SE = sqrt((s1^2/n1) + (s2^2/n2))
SE = sqrt((5.49^2/30) + (5.41^2/30))
Step 4: Determine the critical value corresponding to the desired level of confidence. For a 95% confidence level and two-tailed test, the critical value can be obtained using a t-distribution table with (n1 + n2 - 2) degrees of freedom.
Step 5: Calculate the margin of error (ME), which is the product of the critical value and the standard error:
ME = Critical value * SE
Step 6: Calculate the confidence interval (CI) by subtracting and adding the margin of error from the difference between sample means:
CI = (x1 - x2) +- ME
Plugging in the given values, you can calculate the confidence interval.