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Take a look at our Guided Solution in Statistics!

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

Solution

Given:

Sample size (n) = 100
Sample mean (x) = 10.998
Population standard deviation (σ) = 0.02
Confidence level (1 - α ) = 95% = 0.95

Find:

We need to find the 95% confidence interval for the population mean, μ.

Theoretical Concepts Used in this Solution

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

Confidence level is the percentage of the entire possible samples that are expected to have the population parameter.

Critical Value

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 Error

Standard deviation of the sampling distribution is known as standard error. It is denoted by αx and is given by,

formula6

Where,

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

Where,
Zα ⁄ 2 is critical value
σ x is standard error

Confidence Interval

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)

Where,

x is sample mean

E is margin of error

Steps used for solving:

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.

  1. Critical value, Zα ⁄ 2
  2. Standard error
  3. Margin of error

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,

formula5(Using the normal distribution table)

So, the critical value is 1.96

Standard Error

By substituting the sample size and population standard deviation in the formula

formula4 we get standard error,

Sample size, n = 100

Population standard deviation, σ = 0.02 inches

formula3

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,

formula2

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,

formula1

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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Books on Statistics

Business Statistics, Gupta S.P Operations Management, Joseph G Monks SPSS 13.0 Statistical Procedures Companion, Upper Saddle-River, N.J.Norusis, M. 2004 Probability and Random Variables and Random Signal Principles, Peyton Z Peebles
Statistics For Business And Economics, Anderson Sweeney SAS Institute (Inc)- SAS users guide (version – 9.3.1) Probability and Random Processes, Singaravelu Mathematical Statistics, Kapur J.N.
Applied Logistic Regression, Hosmer, D. W., and S. Lemeshow. 2000. Design and Analysis of Experiments, Monto Merry Elements Of Mathematical Statistics, Gupta.S.C Statistical Design for Research. New York, Kish, L. 987.
Multivariate Data Analysis, Hair Business Statistics, Amir D Aczel Applied Survival Analysis. New York, Hosmer, D. W., and S. Lemeshow. 999 Introduction to Probability Models, Sheldon
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