Suppose the following data are selected randomly from a population of normally distributed values.
42
51 43 48 41 57 54
39 45 48 45 39 46
Construct a 95% confidence interval to estimate the population mean.
Round the intermediate values to 2 decimal places. Round your answers to 2 decimal places, the tolerance is +/-0.05.
I do not understand how to find the 's' for the estimate of standard deviation from the sample.
n=13
xbar=46
df= 12
alpha/2= 0.025
Therefore, t= 2.179
And I am stuck...
To find the estimate of standard deviation from the sample, you can use the formula:
s = √(Σ(xi - x̄)² / (n-1))
where:
- xi represents each individual data point in the sample
- x̄ is the sample mean
- n is the sample size
Applying this formula to your data, we have:
x̄ = (42 + 51 + 43 + 48 + 41 + 57 + 54 + 39 + 45 + 48 + 45 + 39 + 46) / 13 = 47.08 (rounded to 2 decimal places)
Σ(xi - x̄)² = (42 - 47.08)² + (51 - 47.08)² + (43 - 47.08)² + (48 - 47.08)² + (41 - 47.08)² + (57 - 47.08)² + (54 - 47.08)² + (39 - 47.08)² + (45 - 47.08)² + (48 - 47.08)² + (45 - 47.08)² + (39 - 47.08)² + (46 - 47.08)²
= 25.2864 + 13.9104 + 16.1084 + 1.1164 + 38.6304 + 91.4064 + 43.4352 + 60.8764 + 4.3776 + 0.7076 + 4.3776 + 60.8764 + 1.2764
= 361.1224 (rounded to 4 decimal places)
Now we can calculate the estimate of standard deviation:
s = √(Σ(xi - x̄)² / (n-1))
= √(361.1224 / (13-1))
= √(361.1224 / 12)
= √30.09353333
= 5.49 (rounded to 2 decimal places)
Now we have all the necessary values to construct the confidence interval.
To find the estimate of the standard deviation from the sample, you can use the formula for the sample standard deviation:
s = sqrt((∑(xi - xbar)^2) / (n - 1))
Here's the step-by-step process to calculate the estimate of the standard deviation:
1. Calculate the deviations from the sample mean (xbar) for each value in the sample. Subtract the sample mean from each value:
(42 - 46), (51 - 46), (43 - 46), (48 - 46), (41 - 46), (57 - 46), (54 - 46), (39 - 46), (45 - 46), (48 - 46), (45 - 46), (39 - 46), (46 - 46)
2. Square each deviation calculated in step 1:
(-4)^2, (5)^2, (-3)^2, (2)^2, (-5)^2, (11)^2, (8)^2, (-7)^2, (-1)^2, (2)^2, (-1)^2, (-7)^2, (0)^2
3. Calculate the sum of the squared deviations from step 2:
(-4)^2 + (5)^2 + (-3)^2 + (2)^2 + (-5)^2 + (11)^2 + (8)^2 + (-7)^2 + (-1)^2 + (2)^2 + (-1)^2 + (-7)^2 + (0)^2
4. Calculate the sample variance by dividing the sum of squared deviations by (n - 1) (where n is the sample size):
Sample variance = sum of squared deviations / (n - 1)
5. Take the square root of the sample variance to get the estimate of the standard deviation (s).
Now that you have the estimate of the standard deviation (s), you can proceed with constructing the confidence interval to estimate the population mean.
n=13
xbar=46
df= 12
alpha/2= 0.025
Therefore, t= 2.179
Sd = 5.5377
t = 2.179
(42.65, 49.35)
Z = 1.96
(42.99,49.01)