Estimation and Confidence Intervals Lesson 7: Estimation and Confidence Intervals

Outline Point and interval estimates Confidence intervals Student’s t-distribution Degree of freedom Confidence interval for population mean Confidence interval for a population proportion Selecting a sample size Summary

Point and Interval Estimates A point estimate is a single value (statistic) used to estimate a population value (parameter). A confidence interval is a range of values within which the population parameter is expected to occur.

Confidence Intervals The degree to which we can rely on the statistic is as important as the initial calculation. Samples give us estimates of the population parameter – only estimates. Ultimately, we are concerned with the accuracy of the estimate. Confidence interval provides range of values Based on observations from 1 sample Confidence interval gives information about closeness to unknown population parameter Stated in terms of probability Exact closeness not known because knowing exact closeness requires knowing unknown population parameter

Areas Under the Normal Curve If we draw an observation from the normal distributed population, the drawn value is likely (a chance of 68.26%) to lie inside the interval of (µ-1σ, µ+1σ). P((µ-1σ <x<µ+1σ) =0.6826. Between: ± 1  - 68.26% ± 2  - 95.44% ± 3  - 99.74% µ-2σ µ µ+2σ µ-3σ µ-1σ µ+1σ µ+3σ

P(µ-1σ <x<µ+1σ) vs P(x-1σ <µ <x+1σ) P(µ-1σ <x<µ+1σ) is the probability that a randomly drawn observation will lie between (µ-1σ, µ+1σ). P(µ-1σ <x<µ+1σ) = P(µ-1σ -µ-x <x -µ-x<µ +1σ -µ-x) = P(-1σ -x <-µ<1σ -x) = P(-(-1σ -x )>-(-µ)>-(1σ -x)) = P(1σ +x >µ>-1σ +x) = P(x - 1σ <µ <x+1σ) P(x-1σ <µ <x+1σ) is the probability that the population mean will lie between (x-1σ, x+1σ).

P(µ-1σm <m<µ+1 σm) vs P(m-1 σm <µ <m+1 σm) (m=sample mean) P(µ-1 σm <m<µ+1 σm) is the probability that a randomly drawn sample will have a sample mean between (µ-1σ, µ+1σ). P(µ-1 σm <m<µ+1 σm) = P(µ-1 σm -µ-m <x -µ-m<µ +1 σm -µ-m) = P(-1 σm -m <-µ<1 σm -m) = P(-(-1 σm -m )>-(-µ)>-(1 σm -m)) = P(1 σm +m>µ>-1 σm +m) = P(m - 1 σm <µ <m+1 σm) P(m-1 σm <µ <m+1 σm) is the probability that the population mean will lie between (m-1 σm , m+1 σm).

P(µ-a <x<µ+b) vs P(x-a<µ <x+b) P(µ-a <x<µ+b) is the probability that a drawn observation will lie between (µ-a, µ+b). P(x-a <µ <x+b) is the probability that the population mean will lie between (x - a, x+ b). Generally, P(µ-a <x<µ+b) = P(x-a <µ <x+b) NO!!!! Generally, P(µ-a <x<µ+b) and P(x-a <µ <x+b) are not equal. They are equal only if a = b. That is, when the confidence interval is symmetric.

P(µ-a <x<µ+b) = P(x-b <µ <x+a) P(µ-a <x<µ+b) is the probability that a drawn observation will lie between (µ-a, µ+b). P(µ-a <x<µ+b) = P(µ-a -µ-x <x -µ-x<µ +b -µ-x) = P(-a -x <-µ<b -x) = P(-(-a -x )>-(-µ)>-(b -x)) = P(a +x >µ>-b +x) = P(x - b <µ <x+a) P(x-b <µ <x+a) is the probability that the population mean will lie between (x - b, x+ a).

Elements of Confidence Interval Estimation We are concerned about the probability that the population parameter falls somewhere within the interval around the sample statistic. Confidence Interval Confidence Limit (Lower) Confidence Limit (Upper) Sample Statistic Generally, we consider symmetric confidence intervals only.

 x _ Confidence Intervals 90% Samples 95% Samples 99% Samples The likelihood (probability) that the sample mean of a randomly drawn sample will fall within the interval:  x _ 90% Samples 95% Samples 99% Samples

Confidence Intervals The likelihood (or probability) that the sample mean will fall within “1 standard deviation” of the population mean is the same as the likelihood (or probability) that the population mean will fall within “1 standard deviation” of the sample mean. Z 1.645 0.90 0.90 1.96 0.95 0.95 2.58 0.99 0.99

Level of Confidence Probability that the unknown population parameter falls within the interval Denoted (1 -  = level of confidence is the probability that the parameter is not within the interval Typical values are 99%, 95%, 90%

Interpreting Confidence Intervals Once a confidence interval has been constructed, it will either contain the population mean or it will not. For a 95% confidence interval, If we were to draw 1000 samples and construct the 95% confidence interval for the population mean for each of the 1000 samples. Some of the intervals contain the population mean, some not. If the interval is a 95% confidence interval, about 950 of the confidence intervals will contain the population mean. That is, 95% of the samples will contain the population mean. (1-) (1-) (1-) (1-)*1000 (1-)

Intervals & Level of Confidence _ Sampling Distribution of Mean  x  /2  /2 1 -  _ X  =   x Intervals Extend from (1 - ) % of Intervals Contain  .  % Do Not. Large Number of Intervals

Point Estimates and Interval Estimates The factors that determine the width of a confidence interval are: The size of the sample (n) from which the statistic is calculated. The variability in the population, usually estimated by s. The desired level of confidence. =  1 -   /2 X _  x 

Point and Interval Estimates We may use the z distribution if one of the following conditions hold: The population is normal and its standard deviation is known The sample has more than 30 observations (The population standard deviation can be known or unknown). Technical note: If the random variables A and B are normally distributed, Y = A+B and X=(A+B)/2 will be normally distributed. If the population is normal, the sample mean of a random sample of n observations (for any integer n) will be normally distributed.

Point and Interval Estimates Use the t distribution if all of the following conditions are fulfilled: The population is normal The population standard deviation is unknown and the sample has less than 30 observations. Note that the t distribution does not cover those non-normal populations.

The normality assumption in student t Is it possible to check whether the population is normal? Yes. Using normal probability plot and a test of moments. In small samples, however, such check is likely very imprecise. In practice, when sample size is small and the population distribution is known, we use student t. However, if we are very concern whether we should use student t (i.e., the assumption of normality), we might consider some other approach to conduct the inference. One such approach is known as “bootstrap” which is based on simulations. (will be covered only in very advanced statistics course.)

Student’s t-Distribution The t-distribution is a family of distributions that is bell-shaped and symmetric like the standard normal distribution but with greater area in the tails. Each distribution in the t-family is defined by its degrees of freedom. As the degrees of freedom increase, the t-distribution approaches the normal distribution. Student is a pen name for a statistician named William S. Gosset who was not allowed to publish under his real name. Gosset assumed the pseudonym Student for this purpose. Student’s t distribution is not meant to reference anything regarding college students.

Z t Student’s t-Distribution Standard Normal Bell-Shaped Symmetric ‘Fatter’ Tails t (df = 13) t (df = 5) Z t

.05 2 t Student’s t Table  / 2 .05 2.920 t Values Upper Tail Area df Assume: n = 3 df = n - 1 = 2  = .10 /2 =.05  / 2 Upper Tail Area df .25 .10 .05 1 1.000 3.078 6.314 2 0.817 1.886 2.920 .05 3 0.765 1.638 2.353 t 2.920 t Values

Degrees of freedom (df) Degrees of freedom refers to the number of independent data values available to estimate the population’s standard deviation. If k parameters must be estimated before the population’s standard deviation can be calculated from a sample of size n, the degrees of freedom are equal to n - k. Example Sum of 3 numbers is 6 X1 = 1 (or Any Number) X2 = 2 (or Any Number) X3 = 3 (Cannot Vary) Sum = 6 Degrees of freedom = n -1 = 3 -1 = 2

t-Values where: = Sample mean = Population mean s = Sample standard deviation n = Sample size

Confidence interval for mean ( unknown in small sample) A random sample of n = 25 has = 50 and S = 8. Set up a 95% confidence interval estimate for . S S X  t     X  t   / 2 , n  1  / 2 , n  1 n n 8 8 50  2 . 0639     50  2 . 0639  25 25 46 . 69    53 . 30

Constructing General Confidence Intervals for µ In general, a confidence interval for the mean is computed by: Sometimes called Margin of Error (ME).

Margin of Error The confidence interval, Can also be written as where ME is called the margin of error The interval width, w, is equal to twice the margin of error

Reducing the Margin of Error The margin of error can be reduced if the population standard deviation can be reduced (σ↓) The sample size is increased (n↑) The confidence level is decreased, (1 – ) ↓

Finding the Reliability Factor, z/2 Consider a 95% confidence interval: z = -1.96 z = 1.96 Z units: Lower Confidence Limit Upper Confidence Limit X units: Point Estimate Point Estimate Find z.025 = 1.96 from the standard normal distribution table

Common Levels of Confidence Commonly used confidence levels are 90%, 95%, and 99% Confidence Coefficient, Confidence Level Z/2 value 80% 90% 95% 98% 99% 99.8% 99.9% .80 .90 .95 .98 .99 .998 .999 1.28 1.645 1.96 2.33 2.58 3.08 3.27

EXAMPLE 3 The Dean of the Business School wants to estimate the mean number of hours worked per week by students. A sample of 49 students showed a mean of 24 hours with a standard deviation of 4 hours. What is the population mean? The value of the population mean is not known. Our best estimate of this value is the sample mean of 24.0 hours. This value is called a point estimate.

Example 3 continued Find the 95 percent confidence interval for the population mean. The confidence limits range from 22.88 to 25.12. About 95 percent of the similarly constructed intervals include the population parameter.

Confidence Interval for a Population Proportion The confidence interval for a population proportion is estimated by: An biased estimator for the variance of sample proportion. or An unbiased estimator for the variance of sample proportion. The difference of the two estimators is small when n is very large. The textbook uses the former.

EXAMPLE 4 A sample of 500 executives who own their own home revealed 175 planned to sell their homes and retire to Arizona. Develop a 98% confidence interval for the proportion of executives that plan to sell and move to Arizona.

Finite-Population Correction Factor A population that has a fixed upper bound is said to be finite. For a finite population, where the total number of objects is N and the size of the sample is n, the following adjustment is made to the standard errors of the sample means and the proportion: Standard error of the sample means when  is known: Standard error of the sample means when  is NOT known and need to be estimated by s:

Finite-Population Correction Factor Standard error of the sample proportions: or This adjustment is called the finite-population correction factor. If n/N < .05, the finite-population correction factor is ignored.

Selecting a Sample Size There are 3 factors that determine the size of a sample, none of which has any direct relationship to the size of the population. They are: The degree of confidence selected. The maximum allowable error. The variation in the population.

Selecting a Sample Size To find the sample size for a variable: where : E is the allowable error (sometimes called Margin of Error), z is the z- value corresponding to the selected level of confidence, and s is the sample deviation of the pilot survey.

EXAMPLE 6 A consumer group would like to estimate the mean monthly electricity charge for a single family house in July within $5 using a 99 percent level of confidence. Based on similar studies the standard deviation is estimated to be $20.00. How large a sample is required?

Sample Size for Proportions The formula for determining the sample size in the case of a proportion is: where p is the estimated proportion, based on past experience or a pilot survey; z is the z value associated with the degree of confidence selected; E is the maximum allowable error the researcher will tolerate.

EXAMPLE 7 The American Kennel Club wanted to estimate the proportion of children that have a dog as a pet. If the club wanted the estimate to be within 3% of the population proportion, how many children would they need to contact? Assume a 95% level of confidence and that the club estimated that 30% of the children have a dog as a pet. or

Summary: Confidence interval for sample mean General confidence interval: (m = population mean; a = confidence level; s = standard deviation) Sample Size (n) <30 ≥30 s known s unknown Normal Population distribution Unknown ? ?

Summary: Confidence Interval for sample proportion General confidence interval: (p= population mean; a = confidence level; s = standard deviation) Because the variance of one draw from the population is s 2= p(1-p), we know s if only if we know p. If we know p, there is no need to estimate p or to construct the confidence interval for p. Sample Size (n) <30 ≥30 s known s unknown Normal Population distribution Unknown

Lesson 7: Estimation and Confidence Intervals - END -