Answer:
The Confidence Interval = (2180, 2782)
Lower Bound for the true average difference between 1-minute modulus and 4-week modulus = 2180
Upper Bound for the true average difference between 1-minute modulus and 4-week modulus = 2782
- The interval obtained represents the range of values that the true average difference between 1-minute modulus and 4-week modulus can take on.
Step-by-step explanation:
The confidence interval for any distribution is the range of values that the true population mean can take on with some level of confidence.
It is given mathematically as
Confidence Interval = (Sample Mean) ± (Margin of Error)
The Margin of Error is the width of the confidence interval about the mean.
It is given mathematically as
Margin of Error = (Critical Value) × (Standard Error of the Mean)
Before anything, we first state the conditions necessary for the confidence interval calculated to be valid.
- The sample must be a random sample extracted from the population distribution using random sampling technique.
- The sample distribution must be normal or approximately normal and this is confirmed by knowing that the population distribution is normal or almost normal.
These two conditions are assumed to be plausible for this sample distribution.
Here, we need the confidence interval for the true average difference between 1-minute modulus and 4-week modulus.
So, before anything, we need to first fin the sample mean and standard deviation for the differences obtained from the sample specimens.
904, 3370, 2580, 3117, 2850, 2690, 2180, 1241, 2210, 2350, 2260, 2758, 2880, 2750, 3520, 2033
Mean = (Σx/N)
Σx = Sum of all the variables = 39693
N = Sample size = 16
Sample Mean = (39693/16) = 2480.8125
Standard deviation =
x = each variable
xbar = sample mean = 2480.8125
Note that we use (N-1) for the standard deviation because it is a sample standard deviation.
= √(7227548.4375/15) = √(481836.5625)
= 694.14448243863 = 694.1445
Confidence Interval = (Sample Mean) ± (Margin of Error)
Margin of Error = (Critical Value) × (Standard Error of the Mean)
Confidence Interval = (Sample Mean) ± [(Critical Value) × (Standard Error of the Mean)]
Critical Value is obtained from the t-distribution table since no information on the population standard deviation is known.
For that, we need the significance level for the test and the degree of freedom.
Significance level = α = 0.05 (given in the question)
Degree of freedom = N - 1 = 16 - 1 = 15
t(0.05, 15) = 1.753 (from the t-distribution tables)
Standard error of the Mean = σₓ = (σ/) = (694.1445/
) = 173.536125 = 173.536
Confidence Interval = (Sample Mean) ± [(Critical Value) × (Standard Error of the Mean)]
CI = (2480.8125) ± [(1.753 × 173.536)]
CI = (2480.8125) ± (301.09385)
CI = (2179.71865, 2781.90635)
CI = (2180, 2782)
This interval represents the range of values that the true average difference between 1-minute modulus and 4-week modulus can take on.
Hope this Helps!!!