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Sometimes researchers know in advance that they want to estimate a population mean within a specific margin of error for a given level of confidence. In that case, solve the EBM formula for n to discover the size of the sample that is needed to achieve this goal:

n =   z 2 σ 2 E B M 2

Formula review

X ¯ ~ N ( μ X , σ n ) The distribution of sample means is normally distributed with mean equal to the population mean and standard deviation given by the population standard deviation divided by the square root of the sample size.

The general form for a confidence interval for a single population mean, known standard deviation, normal distribution is given by
(lower bound, upper bound) = (point estimate – EBM , point estimate + EBM )
= ( x ¯ E B M , x ¯ + E B M )
= ( x ¯ z σ n , x ¯ + z σ n )

EBM = z σ n = the error bound for the mean, or the margin of error for a single population mean; this formula is used when the population standard deviation is known.

CL = confidence level, or the proportion of confidence intervals created that are expected to contain the true population parameter

α = 1 – CL = the proportion of confidence intervals that will not contain the population parameter

z α 2 = the z -score with the property that the area to the right of the z-score is   2 this is the z -score used in the calculation of "EBM where α = 1 – CL .

n = z 2 σ 2 E B M 2 = the formula used to determine the sample size ( n ) needed to achieve a desired margin of error at a given level of confidence

General form of a confidence interval

(lower value, upper value) = (point estimate−error bound, point estimate + error bound)

To find the error bound when you know the confidence interval

error bound = upper value−point estimate OR error bound = upper value lower value 2

Single Population Mean, Known Standard Deviation, Normal Distribution

Use the Normal Distribution for Means, Population Standard Deviation is Known EBM = z α 2 σ n

The confidence interval has the format ( x ¯ EBM , x ¯ + EBM ).

Use the following information to answer the next five exercises: The standard deviation of the weights of elephants is known to be approximately 15 pounds. We wish to construct a 95% confidence interval for the mean weight of newborn elephant calves. Fifty newborn elephants are weighed. The sample mean is 244 pounds. The sample standard deviation is 11 pounds.

Identify the following:

  1. x ¯ = _____
  2. σ = _____
  3. n = _____
  1. 244
  2. 15
  3. 50

In words, define the random variables X and X ¯ .

Which distribution should you use for this problem?

N ( 244 , 15 50 )

Construct a 95% confidence interval for the population mean weight of newborn elephants. State the confidence interval, sketch the graph, and calculate the error bound.

What will happen to the confidence interval obtained, if 500 newborn elephants are weighed instead of 50? Why?

As the sample size increases, there will be less variability in the mean, so the interval size decreases.


Use the following information to answer the next seven exercises: The U.S. Census Bureau conducts a study to determine the time needed to complete the short form. The Bureau surveys 200 people. The sample mean is 8.2 minutes. There is a known standard deviation of 2.2 minutes. The population distribution is assumed to be normal.

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Source:  OpenStax, Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24. OpenStax CNX. Oct 24, 2015 Download for free at http://legacy.cnx.org/content/col11891/1.8
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