statistics 4 problems IF YOU CANT DO AT 10 DONT WASTE TIME BIDDING I DONT NEED SLOGANS

Chapter08.pptx

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Chapter 8: Sampling Distributions and Estimation

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Population vs. Sample

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Parameter and Statistic:

Parameter:

Descriptive measure of population or distribution

N(μ,σ)

λ for Poisson distribution

Statistic:

Computed from a sample of n items, chosen from a population of N items.

Can be used as estimates of parameters found in the population.

Population vs. Sample

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Point estimate:

Provides a summary of the sample with a single number which is used as an estimate of the population parameter

Example: use to estimate μ or use p to estimate π

Interval estimate:

A range of values within which we believe the population parameter lies with a certain probability (confidence level)

Example: Confidence Interval (CI)

95% Confidence Interval of the mean lies between 502.78 and 504.02

Point vs. Interval Estimate

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Population vs. Sample

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Population vs. Sample

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Estimators and Sampling Distributions

Estimator – a statistic derived from a sample to infer the value of a population parameter.

Estimate – the value of the estimator in a particular sample.

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Examples of Estimators:

Estimators and Sampling Distributions

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Sampling Error:

Comes from the fact that we get different results from sample to sample

It is expected, random and unavoidable

Bias:

is a flaw in the way you are collecting your data.

More serious issue, systematic and avoidable.

Estimators and Sampling Distributions

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Sampling Distributions:

An estimator is a random variable since samples vary.

The sampling distribution of an estimator is the probability distribution of all possible values the estimator may assume when a random sample of size n is taken.

Estimators and Sampling Distributions

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If we repeatedly take samples of size n from a population and calculate the mean ( f ) of each sample; each sample has its own value, and the distribution of these means is called the “sampling distribution of the mean”

Our objective is to use the sampling distribution of the mean ( ) to infer something about the population we are studying

Sampling Distribution of the Mean

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The Central Limit Theorem (CLT) is a powerful result that allows us to approximate the shape of the sampling distribution of the sample mean even when we don’t know what the population looks like.

The Central Limit Theorem (CLT)

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If the population is exactly normal, then the sample mean follows a normal distribution.

The Central Limit Theorem (CLT)

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As the sample size n increases, the distribution of sample means narrows in on the population mean µ.

The Central Limit Theorem (CLT)

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Even if the population is not normal, if the sample size is large enough (n > 30), the sample means will have approximately a normal distribution

The Central Limit Theorem (CLT)

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If the population is symmetric then a sample size less than 30 still ensures that the sample mean follows a normal distribution

The Central Limit Theorem (CLT)

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Confidence Interval

A range of values within which we believe the population parameter lies with a certain probability (confidence level).

If the samples are drawn from a normal population or if the sample is large enough that can be considered normally distributed according to CLT then the confidence interval of the mean can be calculated using the standard normal distribution.

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Confidence Interval

The confidence level (90%, 95%, 99%) is the area under the pdf of the sampling distribution of

The confidence level is equal to 1 – α

Example: if α = 0.1 then the confidence level is 1 – 0.1 = 0.9 (or 90%)

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Confidence Interval for a Mean () with known σ

In this case, the population standard deviation (σ) is given

The confidence interval for mean (μ) with known σ is calculated as follows:

+ ()(

margin of error

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Confidence Interval for a Mean () with known σ

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Some common confidence levels and z-values:

Confidence Interval for a Mean () with known 

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A confidence interval either does or does not contain μ.

The confidence level quantifies the variability of the estimator.

For a 95% CI: out of 100 confidence intervals, approximately 95% may contain μ, while approximately 5% might not contain 

Interpreting Confidence Intervals:

Confidence Interval for a Mean () with known 

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Choosing a Confidence Level:

A higher confidence level leads to a wider confidence interval.

Greater confidence implies loss of precision (i.e. greater margin of error).

Confidence Interval for a Mean () with known 

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Example:

The fuel economy of a car is normally distributed random variable with a standard deviation of 1.25 MPG. A sample of 16 cars has a mean fuel economy of 28 MPG.

Construct a 90% confidence interval for the population fuel economy mean

Confidence Interval for a Mean () with known 

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Is  ever known in the real world?

Yes, but not very often

Example: Manufacturing process

Confidence Interval for a Mean () with known 

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17 = UCL

15 = LCL

16 = Mean

Example

Control Chart for sample of 9 boxes

Sample number

| | | | | | | | | | | |

1 2 3 4 5 6 7 8 9 10 11 12

Variation due to assignable causes

Variation due to natural causes

Out of control

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Use the Student’s t distribution instead of the standard normal distribution when the population is normally distributed but the population standard deviation s is unknown.

Confidence Interval for a Mean () with unknown 

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Student’s t Distribution:

t distributions are symmetric and shaped like the standard normal distribution.

The t distribution is dependent on the size of the sample.

As sample sizes increases, t distribution becomes similar to standard normal distribution

Confidence Interval for a Mean () with unknown 

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Student’s t Distribution:

Confidence Interval for a Mean () with unknown 

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Confidence Interval for a Mean () with unknown 

In this case, the sample standard deviation (s) is given instead of population standard deviation (σ)

The confidence interval for mean (μ) with unknown σ is calculated as follows:

+ ()()

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Confidence Interval for a Mean () with unknown 

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Calculating t statistic:

Degrees of Freedom (d.f.)

Degrees of Freedom (d.f.) is a parameter based on the sample size that is used to determine the value of the

t statistic.

Degrees of freedom tell how many observations are used to calculate s, less the number of intermediate estimates used in the calculation.

Confidence Interval for a Mean () with unknown 

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Calculating t statistic:

Degrees of Freedom (d.f.)

The d.f. for the t distribution is given by: v = n -1

For a given confidence level, t is always larger than z, so a confidence interval based on t is always wider than if z were used.

Confidence Interval for a Mean () with unknown 

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Example: GMAT Scores

Construct a 90% confidence interval for the mean GMAT score of 20 MBA applicants.

x = 510 s = 73.77

Confidence Interval for a Mean () with unknown 

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Comparison of z and t:

For very small samples, t-values differ substantially from the normal

As degrees of freedom increase, the t-values approach the z-values.

For example, for n = 31, d.f. = 31 – 1 = 30

So for a 90 percent confidence interval, we would use

t = 1.697, which is only slightly larger than z = 1.645.

Confidence Interval for a Mean () with unknown 

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Confidence Interval Width:

Confidence interval width reflects: - the sample size - the confidence level - the standard deviation

To obtain a narrower interval and more precision: - increase the sample size or - lower the confidence level (e.g., from 90% to 80% confidence).

Confidence Interval for a Mean () with unknown 

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Using Appendix D:

Beyond d.f. = 50, Appendix D shows d.f. in steps of 5 or 10.

If the table does not give the exact degrees of freedom, use the t-value for the next lower degrees of freedom.

This is a conservative procedure since it causes the interval to be slightly wider.

Confidence Interval for a Mean () with unknown 

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Student’s t distribution

If X follows the t-distribution, we can calculate

P(X < x) or P(X > x) using the t-table

Example:

Assume X follows a t distribution with 12 degrees of freedom, find the following:

P(X < 1.782)

P(X > -1.356)

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Student’s t distribution

If X follows the t-distribution, we can calculate k in

P(X< k) = m or P(X > k) = m using the t-table

Example:

Assuming X follows a t distribution, find k values:

P(X > k) = 0.05 where v = 18

P(X < k) = 0.90 where v = 22

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Student’s t distribution

T-transform:

t =

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The Central Limit Theorem (CLT) also applies to a sample proportion.

Central Limit Theorem for a Proportion

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The distribution of a sample proportion p = is symmetric if

π = .50 and regardless of π, approaches symmetry as n increases.

Central Limit Theorem for a Proportion

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When is it Safe to Assume Normality of p?

Rule of Thumb: The sample proportion p = may be assumed to be normal if both np 10 and n(1- p) 10.

Sample size to assume normality:

Table 8.9

Central Limit Theorem for a Proportion

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Confidence Interval for π:

The confidence interval for π is

Confidence Interval for a Proportion ()

p + (

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Example: Auditing

A sample of 75 retail in-store purchases showed that 24 were paid in cash. Construct a 95% confidence interval for the proportion of all retail in-store purchase that are paid by cash.

Confidence Interval for a Proportion ()

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Let Y be a random variable from a standard normal distribution

Y ~ N(0,1)

Let Q be a random variable which is the square of random variable Y

Q = Y2

The distribution of Q is chi-square

Q ~ 2(k) with k degrees of freedom

where k = n-1

Chi-Square Distribution

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Properties:

Not symmetric

Shape depends on degrees of freedom

As the number of degrees of freedom increase, the distribution becomes more symmetric

Chi-Square Distribution

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Example:

Assume X follows a chi-square distribution with 14 degrees

of freedom. Using the chi-distribution table, find the following probabilities:

P(X > 21.06)

P(X < 6.571)

Find x such that P(X > x) = 0.9

Chi-Square Distribution

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If the population is normal, then the sample variance s2 follows the chi-square distribution (2) with degrees of freedom d.f. = n – 1

2 Transform:

Chi-Square Distribution

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Confidence Interval:

Using the sample variance s2, the confidence interval is

Chi-square distribution table is used to obtain and

To obtain a confidence interval for the standard deviation , just take the square root of the interval bounds.

Confidence Interval for a Population Variance (2)

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Confidence Interval:

Confidence Interval for a Population Variance (2)

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Example:

A pediatrician’s records showed the mean height of a random sample of 25 girls at age 12 months to be 29.53 inches with a standard deviation of 1.095. Construct a 95% confidence interval for the population variance assuming the population is normally distributed.

Confidence Interval for a Population Variance (2)

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