Find the 95% confidence interval for the difference between two means based on this information about two samples. Assume independent samples from normal populations. (Use conservative degrees of freedom.) (Give your answers correct to two decimal places.)
Sample Number Mean Std. Dev.
1 27 39 21
2 16 26 35
Lower Limit
Upper Limit
xbar1 - xbar2 -+ ta/2 * sqrt(s1^2/n2 + s2^2/n2)
13 -+ 2.02* sqrt(21^2/27 + 35^2 /16))
(-6.47, 32.47)
Those was not the right answers, the ones that I had got previous was also close to them.
For pooled
sp = (n1-1)s1^2 + (n2-1)s2^2 /(n1+n2-2)
Sqrt(29841/41 )
= 26.978
Confidence interval
The degrees of freedom of t is n1+ n2 -2
xbar1-xbar2 -+ta/2 *sp*sqrt(1/n1 +1/n2))
(39-26)-+ 2.02* 26.978sqrt(1/27+ 1/16))
(-4.19, 30.19)
Thanks, but those are wrong also.
conservative degrees of freedom, you use small number. Df = 15
(39-26)-+ 2.13* sqrt(21^2/27 + 35^2/16))
13 -+ 2.13*sqrt(4459/48)
13 -+ 20.53
(-7.53, 33.53)
To find the 95% confidence interval for the difference between two means, we need to calculate the mean difference and the margin of error. Here's how you can do it step by step:
Step 1: Calculate the mean difference (D) between the two samples.
D = mean of Sample 1 - mean of Sample 2
D = 27 - 16 = 11
Step 2: Calculate the standard deviation of the difference (SD).
SD = sqrt((std. dev. of Sample 1)^2/n1 + (std. dev. of Sample 2)^2/n2)
SD = sqrt((39^2/21) + (26^2/35)) = sqrt(1521/21 + 676/35) = sqrt(72.43 + 19.31) = sqrt(91.74) ≈ 9.58
Step 3: Calculate the margin of error (ME).
ME = Critical value * SD
To find the critical value, we need to determine the degrees of freedom (df) for this particular situation. Since the conservative degrees of freedom should be used, we use the smaller value between (n1 - 1) and (n2 - 1).
df = min(n1 - 1, n2 - 1) = min(21 - 1, 35 - 1) = min(20, 34) = 20
Using a t-distribution table with a confidence level of 95% and 20 degrees of freedom, the critical value is approximately 2.086.
ME = 2.086 * 9.58 = 19.96
Step 4: Calculate the lower and upper limits of the confidence interval.
Lower Limit = D - ME = 11 - 19.96 = -8.96
Upper Limit = D + ME = 11 + 19.96 = 30.96
Therefore, the 95% confidence interval for the difference between the two means is (-8.96, 30.96).