# 8.1 A single population mean using the normal distribution  (Page 7/20)

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“Metadata Description of Candidate Summary File.” U.S. Federal Election Commission. Available online at http://www.fec.gov/finance/disclosure/metadata/metadataforcandidatesummary.shtml (accessed July 2, 2013).

“National Health and Nutrition Examination Survey.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/nchs/nhanes.htm (accessed July 2, 2013).

## Chapter review

In this module, we learned how to calculate the confidence interval for a single population mean where the population standard deviation is known. When estimating a population mean, the margin of error is called the error bound for a population mean ( EBM ). A confidence interval has the general form:

(lower bound, upper bound) = (point estimate – EBM , point estimate + EBM )

The calculation of EBM depends on the size of the sample and the level of confidence desired. The confidence level is the percent of all possible samples that can be expected to include the true population parameter. As the confidence level increases, the corresponding EBM increases as well. As the sample size increases, the EBM decreases. By the central limit theorem,

$EBM=z\frac{\sigma }{\sqrt{n}}$

Given a confidence interval, you can work backwards to find the error bound ( EBM ) or the sample mean. To find the error bound, find the difference of the upper bound of the interval and the mean. If you do not know the sample mean, you can find the error bound by calculating half the difference of the upper and lower bounds. To find the sample mean given a confidence interval, find the difference of the upper bound and the error bound. If the error bound is unknown, then average the upper and lower bounds of the confidence interval to find the sample mean.

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:

## Formula review

$\overline{X}~N\left({\mu }_{X},\frac{\sigma }{\sqrt{n}}\right)$ 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 )
= $\left(\overline{x}-EBM,\overline{x}+EBM\right)$
= $\left(\overline{x}-z\frac{\sigma }{\sqrt{n}},\overline{x}+z\frac{\sigma }{\sqrt{n}}\right)$

EBM = $z\frac{\sigma }{\sqrt{n}}$ = 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

${z}_{\frac{\alpha }{2}}$ = the z -score with the property that the area to the right of the z-score is this is the z -score used in the calculation of "EBM where α = 1 – CL .

n = $\frac{{z}^{2}{\sigma }^{2}}{EB{M}^{2}}$ = the formula used to determine the sample size ( n ) needed to achieve a desired margin of error at a given level of confidence

#### Questions & Answers

Let x1, x2, ...,xn be a random sample of size n from N(0,σ  ), show that there exists an UMP test with significance level α for testing H0 :  2 =  2 against H1 :  2 <  2 . If n=15,  = 0.05, and  2= 3, determine the BCR
Bhavana Reply
3xy^2√[x^3y^2/(12(x^3y)^2)]
Esther Reply
what is probability
Esther
what is probability
Esther
what is probability
Esther
Probability is a branch of mathematics that deals with the occurrence of a random event. For example, when a coin is tossed in the air, the possible outcomes are Head and Tail.
Dewan
mean 0 and standard deviation 1 .using area table find P(X>3)
Naeem Reply
what is terms data?
Mohsin Reply
define the types of data?
Mohsin
define the number of classes?
Mohsin
define the class limt?
Mohsin
define the class frequency and class interval ?
Mohsin
define class boundaries
George
Your home address nominal Interval ratio ordinal
MD
home address is nominal
Awel
what is axioms of probability
Khaleed Reply
characteristics of statistic
Safi Reply
a measure of cntral tendency is a quantitative value that tends to locate in some sense the middle of a set of data
khan Reply
in a large metropolitan area
Vernon Reply
a lecturer claims that his students score an average of 55 marks in their statistics test. the object supervisor wants to know wether the lecturer's claims is acceptable or not.what is the posible tail of the test?
Karen
2
J-zil
the best sampling method for A school has a total of 100 teachers. Each teacher in the school is given a number and then a random sample of 35 teachers is obtained.
Nurhaznissah Reply
what the best sampling method for A school has a total of 100 teachers. Each teacher in the school is given a number and then a random sample of 35 teachers is obtained.
Nurhaznissah
option please
Dewan
systematic
Dewan
any one send me the notes of these chpt if possible introduction to statistics measure of centeral tendency or average measure of dispensation moments and skewness presentation of data
Aqsa Reply
Kindly send me these notes.
Naheed
what is a regression, and what is it primarily used for
Denisha Reply
assume the sample populations do not have equal standard deviations and use the 0.05 significance level
Nokuthula Reply
what is the solution to this question?
Manbyen Reply
hi
Dewan
hello
Learn
hi please tell
Dewan
The controls that are usually used are
Rushikesh Reply
what is math
Rushikesh
the controls that are usually used in quality controls and also controls a process is key tool used in run chat, control chat and design of experiment etc.,
Sravanthi
mean is number that occurs frequently in a giving data
Chinedu Reply
That places the mode and the mean as the same thing. I'd define the mean as the ratio of the total sum of variables to the variable count, and it assigns the variables a similar value across the board.
Samsicker

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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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