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Find the 95% confidence interval for population mean (μ) using the following information.
Population mean: 11
Standard deviation: 0.02
Random sample: 100
Sample mean: 10.998
Sample size (n) = 100
Sample mean (x) = 10.998
Population standard deviation (σ) = 0.02
Confidence level (1 - α ) = 95% = 0.95
We need to find the 95% confidence interval for the population mean, μ.
Interval estimates are based on sampling distributions. According to central limit theorem, the sample mean is normally disitributed with mean µx and variance σ2x. Since population standard deviation is given, we know that the distribution is normal. So, the formulae presented are related to normal distribution. For other distributions the critical value and standard error will differ depending on the specific distribution.
For solving the problem it is important to know the following important notions and formulae.
Confidence level is the percentage of the entire possible samples that are expected to have the population parameter.
Critical value is a factor that is used to calculate margin of error. Critical value, Zα ⁄ 2, is found using the normal distribution table.
Standard deviation of the sampling distribution is known as standard error. It is denoted by αx and is given by,
σ = population standard deviation n = sample size
Margin of Error
The range of values that is higher and lower than the sample statistic is known as margin of error. It actually indicates the sampling error. Margin of error is denoted by E and the formula for margin of error is,
E = (Critical Value) * (Standard Error)
E = Zα ⁄ 2 * σ x
Zα ⁄ 2 is critical value
σ x is standard error
Confidence interval is a range of sample statistic that contains a population parameter. The confidence level indicates the percentage upto which the interval contains the true population parameter.
Confidence interval is given by, (x — E, x + E)
x is sample mean
E is margin of error
Let us find the 95% confidence interval for the population mean μ. We will proceed step by step. In order to find the confidence interval, we have to compute the following.
Critical Value, Zα ⁄ 2
Let us first find the critical value. We know that confidence level is given by,
1 - α = 0.95 α = 1-0.95 α = 0.05
Dividing both sides by 2 we get,
(Using the normal distribution table)
So, the critical value is 1.96
By substituting the sample size and population standard deviation in the formula
we get standard error,
Sample size, n = 100
Population standard deviation, σ = 0.02 inches
Standard error, S.E = 0.002
Margin of Error
Margin error is nothing but the product of critical value and standard error. Therefore, margin of error is,
Margin of error, E = 0.00392
The formula for confidence interval is (x - E, x + E)
By substituting x = 10.998 and E = 0.00392 in the above formula, we get,
Therefore, the 95% confidence interval for the population mean μ is (10.99408,11.00192) and we are 95% confident that the population mean (μ) is contained in the interval (10.99408,11.00192).
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