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To construct a confidence interval for the mean, you will need: a. a point estimate. b. the expected value from the underlying distribution. c. the degrees of freedom. d. a p
Answer to: To construct a confidence interval for the mean, you will need: a. a point estimate. b. the expected value from the underlying...
To construct a confidence interval for the mean, you will need: a. a point estimate. b. the...
To construct a confidence interval for the mean, you will need: a. a point estimate. b. the... Question:
To construct a confidence interval for the mean, you will need:
a. a point estimate.
b. the expected value from the underlying distribution.
c. the degrees of freedom.
d. a p-value.
The estimated interval (for known population standard deviation), for a provided confidence level, is given as,
¯¯¯ x ± z σ √ n x¯±zσn
For an unknown population standard deviation, the true mean interval is assumed as,
¯¯¯ x ± t s √ n x¯±tsn
, where s=sample standard deviation
The confidence interval will be estimated with a less margin of error, as the sample size is increased. And the point estimate (i.e sample mean in this case) is to be calculated from the considered sample.
Answer and Explanation:
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As the sample mean is representative of the true mean, hence it is necessary to be evaluated. The margin of order is subtracted and added with the...
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Calculating Confidence Intervals, Levels & Coefficients
Chapter 9 / Lesson 2
Confidence intervals demonstrate how sure researchers are that a mean will lie between two numbers. Identify the importance of point and interval estimates, confident levels and coefficients, and terms associated with confidence in statistics.
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Confidence Interval for the Mean
How to Calculate a Confidence Interval for a Population Mean When You Know Its Standard Deviation
You've got the standard deviation. Now you want to figure out a confidence interval for the average of a population. Find out how.
How to Calculate a Confidence Interval for a Population Mean When You Know Its Standard DeviationBy: Deborah J. RumseyUpdated: 03-15-2022
If you know the standard deviation for a population, then you can calculate a confidence interval (CI) for the mean, or average, of that population. When a statistical characteristic that’s being measured (such as income, IQ, price, height, quantity, or weight) is most people want to estimate the mean (average) value for the population. You estimate the population mean, μ, by using a sample mean, x̄, plus or minus a margin of error. The result is called a
When the population standard deviation is known, the formula for a confidence interval (CI) for a population mean is x̄ ± z* σ/√n, where x̄ is the sample mean, σ is the population standard deviation, n is the sample size, and represents the appropriate *-value from the standard normal distribution for your desired confidence level.
*values for Various Confidence Levels
Confidence Level z*-value
90% 1.645 (by convention)
95% 1.96 98% 2.33 99% 2.58
The above table shows values of for the given confidence levels. Note that these values are taken from the standard normal (Z-) distribution. The area between each z* value and the negative of that z* value is the confidence percentage (approximately). For example, the area between z*=1.28 and z=-1.28 is approximately 0.80. Hence this chart can be expanded to other confidence percentages as well. The chart shows only the confidence percentages most commonly used.
In this case, the data either have to come from a normal distribution, or if not, then has to be large enough (at least 30 or so) in order for the Central Limit Theorem to be applied, allowing you to use values in the formula.
To calculate a CI for the population mean (average), under these conditions, do the following:
Determine the confidence level and find the appropriate -value.
Refer to the above table.
Find the sample mean (x̄) for the sample size ().The population standard deviation is assumed to be a known value, σ.
Multiply times σ and divide that by the square root of .
This calculation gives you the margin of error.
Take x̄ plus or minus the margin of error to obtain the CI.
The lower end of the CI is x̄ minus the margin of error, whereas the upper end of the CI is x̄ plus the margin of error.
For example, suppose you work for the Department of Natural Resources and you want to estimate, with 95 percent confidence, the mean (average) length of all walleye fingerlings in a fish hatchery pond.
Because you want a 95 percent confidence interval, your -value is 1.96.
Suppose you take a random sample of 100 fingerlings and determine that the average length is 7.5 inches; assume the population standard deviation is 2.3 inches. This means x̄ = 7.5, σ = 2.3, and n = 100.
Multiply 1.96 times 2.3 divided by the square root of 100 (which is 10). The margin of error is, therefore, ± 1.96(2.3/10) = 1.96*0.23 = 0.45 inches.
Your 95 percent confidence interval for the mean length of walleye fingerlings in this fish hatchery pond is 7.5 inches ± 0.45 inches.
(The lower end of the interval is 7.5 – 0.45 = 7.05 inches; the upper end is 7.5 + 0.45 = 7.95 inches.)
After you calculate a confidence interval, make sure you always interpret it in words a non-statistician would understand. That is, talk about the results in terms of what the person in the problem is trying to find out — statisticians call this interpreting the results “in the context of the problem.”
In this example you can say: “With 95 percent confidence, the average length of walleye fingerlings in this entire fish hatchery pond is between 7.05 and 7.95 inches, based on my sample data.” (Always be sure to include appropriate units.)
About This Article
This article is from the book:
Statistics For Dummies, 2nd Edition
About the book author:
Deborah Rumsey, PhD, is an auxiliary faculty member and program specialist in department of statistics at The Ohio State University. An author of several Dummies books, she is a fellow of the American Statistical Association.
This article can be found in the category:
Statistics For Dummies Cheat Sheet
Checking Out Statistical Confidence Interval Critical Values
Handling Statistical Hypothesis Tests
Statistically Figuring Sample Size
Surveying Statistical Confidence Intervals
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