Point and Interval Estimation

Point and Interval Estimation

Presented by: Shubham Mehta 0019

GOALS

1. Methods of Estimation2. Difference between Point and Interval Estimation3. Defining Level of Confidence4. Constructing Confidence Intervals 5. Interpretation of these confidence intervals6. Determine the sample size for attribute and

variable sampling.7. Explanations using examples /case study

What are Estimators?

In statistics, an estimator is a function of the data or sample that is used to infer the value of an unknown parameter in population in a statistical model.

Thus the estimator, the quantity of interest (the estimand or parameter) and its result (the estimate) are different from each other.

Qualities of Estimators…Statisticians have already determined the “best” way to estimate a population

parameter. Qualities desirable in estimators include

unbiasedness, consistency, and relative efficiency:• An unbiased estimator of a population parameter

is an estimator whose expected value is equal to that parameter.

• An unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows smaller as the sample size grows larger.

• If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to be relatively efficient.

Estimation…

There are two types of inference: estimation and hypothesis testing; estimation is introduced first.

The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic.

E.g., the sample mean ( ) is employed to estimate the population mean ( ).

Point & Interval Estimation…

For example, suppose we want to estimate the mean summer income of a class of business students. For n=25 students.

It is calculated and average is found to be 400 $/week.

point estimate interval estimate

An alternative statement is: The mean income is between 380 and 420 $/week.

10.6

Estimation…

The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic.

There are two types of estimators:

Point Estimator

Interval Estimator

Point Estimator… A point estimator draws inferences about a population

by estimating the value of an unknown parameter using a single value or point.

We saw earlier that point probabilities in continuous distributions were virtually zero. Likewise, we’d expect that the point estimator gets closer to the parameter value with an increased sample size, but point estimators don’t reflect the effects of larger sample sizes. Hence we will employ the interval estimator to estimate population parameters…

Interval Estimator…

An interval estimator draws inferences about a population by estimating the value of an unknown parameter using an interval.

That is we say (with some ___% certainty) that the population parameter of interest is between some lower and upper bounds.

5. Statistical Inference: Estimation

Goal: How can we use sample data to estimate values of population parameters?

Point estimate: A single statistic value that is the “best guess” for the parameter value

Interval estimate: An interval of numbers around the point estimate, that has a fixed “confidence level” of containing the parameter value. Called a confidence interval.

(Based on sampling distribution of the point estimate)

Point Estimators – Most common to use sample values

Sample mean estimates population mean m

ˆ iyyn

m

• Sample std. dev. estimates population std. dev. s

2( )ˆ

1iy y

sn

s

• Sample proportion p estimates population proportion p^.

Methods of finding estimators…

Point estimators Method of moment

Maximum likelihood estimators

Bayes estimators

The EM algorithm

Interval estimators Inverting a test statistic

Pivotal quantities

Pivoting the CDF

Bayesian intervals

Confidence IntervalConfidence Interval of a parameter consists of an interval of

numbers along with a probability that the interval contains the unknown parameter

A confidence interval gives a range estimate of values: Takes into consideration variation in sample statistics from sample to

sample Based on all the observations from one sample Gives information about closeness to unknown population parameters Stated in terms of level of confidence Example: 95% confidence, 99% confidence

Can never be 100% confident A confidence interval estimate of 100% would be so

wide as to be meaningless for practical decision making

Confidence IntervalThe larger the CI , the more confident we can be that

the given interval contains the unknown parameter. Ideally, we prefer a short interval with high degree of

confidence.

For Example: We will prefer (95,100) with 95% confidence than (0,100) with 100% confidence.

The value corresponding to a significance level that determines those test statistics that lead to rejection of null hypothesis and those that lead to a decision not to reject null hypothesis is referred to as Critical Value.

The general formula for all confidence intervals is:

Point Estimate ± (Critical Value) (Standard Error)

Sample Meanor

Sample ProportionThe “z” or “t”Critical Value

σ / √ n or s / √ n

Confidence Interval Estimates

The Level of Significance (α)

Because we only select one sample, and μ or π

are unknown, we never really know whether the

confidence interval includes the true population

mean or proportion, or not. The level of significance, or “α” risk is the

chance we take that the true population parameter

is not contained in the confidence interval.

Therefore, a 95% confidence interval would have

an “α” of 5%

95% Confidence Interval

“ α “ is the proportion in the tails of the sampling distributionthat is outside the established confidence interval.

If α = .05, then each tail has

.025 area

a = .025a = .025

.9750.0250

+ 1.96 z- 1.96 z

Z .06

- 1.9 .0250

Z .06

+ 1.9 .9750

The critical values of “z” thatdefine the “α” areas are

-1.96 and + 1.96

Point Estimate

The Level of Significance (α)

Level of Confidence

The Level of Confidence in a confidence interval is a probability that represents the percentage of intervals that will contain if a large number of repeated samples are obtained. The level of confidence is denoted

For example, a 95% level of confidence would mean that if 100 confidence intervals were constructed, each based on a different sample from the same population, we would expect 95 of the intervals to contain the population mean.

Constructing Confidence Interval

The construction of a confidence interval for the population mean depends upon three factors

1. The point estimate of the population2. The level of confidence 3. The standard deviation of the sample mean

1. Select our desired level of confidence Let’s suppose we want to construct an

interval using the 95% confidence level2. Calculate α and α/2 (1-α)*100% = 95% α = 0.05, α/2 = 0.0253. Look up the corresponding z-score α/2 = 0.025 a z-score of 1.96


4. Multiply the z-score by the standard error to find the margin of error

5. Find the interval by adding and subtracting this product from the mean

).,.( 2/2/ errorstdZxerrorstdZx

nnZ ss 96.12/

)96.1,96.1(n

xn

x ss


95% Confidence Interval

.9750.0250

+ 1.96 z- 1.96 z

Point Estimate


a = .025a = .025

Graph will be as follows

Common Confidence Levels and α values

Here is a table of commonly used confidence levels, α and α/2 values, and corresponding z-scores which we are using in our examples:

• (1 - α)*100% α α/2 Zα/2• 90% 0.1 0.05 1.645• 95% 0.05 0.025 1.96• 99% 0.01 0.005 2.58

Ex : Suppose we conduct a poll to try and get a sense of the outcome of an upcoming election with two candidates. We poll 1000 people, and 550 of them respond that they will vote for candidate A

How confident can we be that a given person will cast their vote for candidate A?

1. Select our desired levels of confidence We’re going to use the 90%, 95%, and 99% levels

Constructing Confidence IntervalExample

2. Calculate α and α/2

Our values are 0.1, 0.05, and 0.01 respectively

Our /2 values are 0.05, 0.025, and 0.005

3. Look up the corresponding z-scores

Our Z/2 values are 1.645, 1.96, and 2.58

4. Multiply the z-score by the standard error to find the margin of error

First we need to calculate the standard error


5. Find the interval by adding and subtracting this product from the mean

In this case, we are working with a distribution we have not previously discussed, a normal binomial distribution (i.e. a vote can choose Candidate A or B, a binomial function)

We have a probability estimator from our sample, where the probability of an individual in our sample voting for candidate A was found to be 550/1000 or 0.55

We can use this information in a formula to estimate the standard error for such a distribution:


5. Multiply the z-score by the standard error cont.• For a normal binominal distribution, the standard

error can be estimated using:

sX =s

n =(p)(1-p)

n=

(0.55)(0.45)1000

= 0.0157

• We can now multiply this value by the z-scores to calculate the margins of error for each conf. level


5. Multiply the z-score by the standard error cont.• We calculate the margin of error and add and subtract that

value from the mean (0.55 in this case) to find the bounds of our confidence intervals at each level of confidence:

Margin Bounds CI Z/2 of error Lower Upper

90% 1.645 0.026 0.524 0.57695% 1.96 0.031 0.519 0.58199% 2.58 0.041 0.509 0.591


Some MythsWhat if we can make a“guess”about proportion value?How can we predict whether it will rain tomorrow?Be Careful! The following statement is NOT true:

“The probability that µ lies between 143.22 and 162.78 is .95.”

Once you have inserted your sample results into the confidence interval formula, the word PROBABILITY can no longer be used to describe the resulting confidence interval.

Some comments about CI’S

Effects of n, confidence coefficient true for CIs for other parameters also

If we repeatedly took random samples of some fixed size n and each time calculated a 95% CI, in the long run about 95% of the CI’s would contain the population proportion .

The probability that the CI does not contain is called the error probability, and is denoted by α.

α = 1 – confidence coefficient

Comments about CI for population mean µ

The method is robust to violations of the assumption of a normal population distribution

(Be careful if sample data distribution is very highly skewed, or if it contains severe outliers)

Greater confidence requires wider CIGreater n produces narrower CI

To determine sample size:

Know the desired confidence level, which determines the value of Z (the critical value from the standardized normal distribution. Determining the confidence level is subjective.

Know the acceptable sampling error, e. The amount of error that can be tolerated.

Know the standard deviation, σ. If unknown, estimate by past data or make an educated guess

estimate σ: [σ = range/4] This estimate is derived from the empirical rule stating that approximately 95% of the values in a normal distribution are within +/- 2σ of the mean, giving a range within which most of the values are located.

Selecting the Sample Size…

We can control the width of the interval by determining the sample size necessary to produce narrow intervals.Suppose we want to estimate the mean demand “to within 5 units”; i.e. we want to the interval estimate to be:

Since:

It follows that Solve for n to get requisite sample size!

Selecting the Sample Size…

The amount of sampling error you are willing to accept and the level of confidence desired, determines the size of your sample.

n = Z2σ2 / e2

e = Z (σ / √ n )

Choosing the Sample Size

Ex. How large a sample size do we need to estimate a population proportion (e.g., “very happy”) to be within 0.03, with probability 0.95?

i.e., what is n so that margin of error of 95% confidence interval is 0.03?

Set 0.03 = margin of error and solve for n

ˆ0.03 1.96 1.96 (1 ) / ns

Some comments about CIs and sample size

We’ve seen that n depends on confidence level (higher confidence requires larger n) and the population variability (more variability requires larger n)

In practice, determining n not so easy, because (1) many parameters are to be estimated, (2) resources may be limited and we may need to compromise, due to several constraints.

CI’s can be formed for any parameter (i.e., for median, mode etc.)

How large must a sample be for the Central Limit theorem to apply?

The sample size varies according to the shape of the population. However, for our use, a sample size of 30 or larger will suffice.

Q. Must sample sizes be 30 or larger for populations that are normally distributed?

Ans: No. If the population is normally distributed, the sample means are normally distributed for sample sizes as small as n=1.

Q. How large is large?Q. Why not just always pick a sample size of 30?

Aim: Finding an optimum size for the sample

How can I tell the shape of the underlying population?

CHECK FOR NORMALITY: Use descriptive statistics. Construct stem-and-leaf plots

for small or moderate-sized data sets and frequency distributions and histograms for large data sets.

Compute measures of central tendency (mean and median) and compare with the theoretical and practical properties of the normal distribution. Compute the interquartile range. Does it approximate the 1.33 times the standard deviation?

How are the observations in the data set distributed? Do approximately two thirds of the observations lie between the mean and plus or minus 1 standard deviation? Do approximately four-fifths of the observations lie between the mean and plus or minus 1.28 standard deviations? Do approximately 19 out of every 20 observations lie between the mean and plus or minus 2 standard deviations?

Interpreting a Confidence Interval

For the previous 95% confidence interval, the following conclusions are valid: I am 95% confident that the average length of a call for

the population µ, lies between 143.22 and 162.78 minutes.

If I repeatedly obtained samples of size 85, then 95% of the resulting confidence intervals would contain µ and 5% would not.

QUESTION: Does this confidence interval [143.22 to 162.78] contain µ?

ANSWER: I don’t know. All I can say is that this procedure leads to an interval containing µ 95% of the time.

I am 95% confident that my estimate of µ [namely 153 minutes] is within 9.78 minutes of the actual value of µ. RECALL: 9.78 is the margin of error.

Interpretations (contd.)

Therefore, 7.93% of the time, a random sample of 40 customers from this population will yield a mean expenditure of 8700 or more.

OR

From any random sample of 40 customers, 7.93% of them will spend on average 8700 or more.

Interpretations (contd.)

The point estimate for this problem is 13.56 hours, with an error of +/- 3.20 hours.

I am 90% confident that the average amount of time accumulated by a manager per week in this industry is between 10.36 and 16.76 hours.

We are 95% confident that the population proportion of telemarketing firms that use their operation to assist order processing is somewhere between .29 and .49.

There is a point estimate of .39 with a margin of error of +/- .10.

Assumptions necessary to use t-distribution

Assumes random variable x is normally distributed

However, if sample size is large enough ( > 30), t-distribution can be used when σ is unknown.

But if sample size is small, evaluate the shape of the sample data using a histogram or stem-and-leaf.

As the sample size increases, the t-distribution approaches the Z distribution.

Statistical Inference facilitates decision making.

Point and Interval Estimation

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