Statistical Inference

ByDr. Manas Kumar Pal

Statistics

It is a branch of mathematics used to summarize, analyze & interpret a group of numbers of observations.

Types of Statistics• Descriptive Statistics :

It summarize data to make sense or meaning of a list of numeric values.• Inferential Statistics :

It is used to infer or generalize observations made with samples to the larger population from which they were selected. Broadly it is classified into theory of estimation and testing of hypothesis

Estimation & Testing of Hypothesis

EstimationThe method to estimate the value of a population parameter from the value of the corresponding sample statistic.

Testing of HypothesisA claim or belief about an unknown parameter value.

Types of Estimation

• Point estimationIt is the value of sample statistic that is used to estimate most likely value of the unknown population parameter.

Methods of point estimationMethod of maximum likelihoodMethod of least squaresMethod of moments

• Interval estimationIt is the range of the values that is likely to have population parameter value with a specified level of confidence.

Properties of estimation• Consistency

The statistic tend to become closer to population parameter as the sample size increases.

• UnbiasednessE(Statistic) = Parameter

• EfficiencyRefers to the size of the standard error(SE). E.g., SE of sample median is greater than the sample mean, So the sample mean is more efficient .

• SufficiencyRefers to the usage of sample information by the statistic. E.g., Sample mean is more sufficient than sample median because usage is more.

Drawback of point estimation

No information is available regarding its reliability i.e., how close it is to its true population parameter.

In fact, the probability that a single sample statistic actually equals to the population parameter is extremely small

Interval Estimation

Confidence Interval= Point estimate ± margin of error

Margin of error = (critical value of ‘Z’ or ‘t’ at 90%, 95% & so on confidence level) x (standard error of particular statistic)

Estimation

Population mean – Avg. salary Population proportion – Stock Market

Interval Estimation for population mean(µ)

SAMPLE SIZELarge Sample(n≥30)

• Known SD(σ)

• Unknown SD(σ)

• Sample Mean square(S)

FORMULAE

nSZx

2

nZx

2

211

xxn

Interval Estimation for population mean(µ)

SAMPLE SIZESmall Sample(n<30)

• Known SD(σ)

• Unknown SD(σ)

• Sample Mean square(S)

FORMULAE

nStx

2

nZx

2

211

xxn

Interval estimation for population proportion(P)

nPPZpP )1(

2

nppZpP )1(

2

If population proportion is given

If population proportion is not given

1. A random sample of size 20 is drawn from a normal population with mean 28 and variance 25 has a sample mean 30. What is the 95% confidence interval?

2. A random sample 50 pieces of certain cord was tested and the mean breaking strength is found to be 15.6 kgs and standard deviation of 2.2 kgs. Use 1% level of significance & to find confidence interval.

3. A cable TV operator claims that 45 % of the homes in a city have opted for his services. Before sponsoring advertisements on the local cable channel, a company conducted a survey & found that 200 out of 550 persons were found to have cable TV services from the operator . Set up confidence interval at 5% level of significance.

4. A departmental store wants to determine the percentage of shoppers who buy at least one of them. A random sample of 5oo shoppers leaving the shop showed that 150 did not buy any item. What is the 90% confidence interval for the percentage of buyers?

PROBLEM ON ESTMATION

5. A manufacturer of computer paper has a production process that operates continuously throughout an entire production shift. The paper is expected to have a mean length 11 inches and the standard deviation of length known to be 0.02 inch. At periodic intervals, samples are selected to determine whether the mean paper length is still equal to 11 inches or something has gone wrong in the production process to change the length of the paper produced. If such a situation has occurred, corrective action is needed. Suppose a random sample of 100 sheets is selected. And the mean paper length is found to be 10.998 inches. Set up 95% and 99% confidence interval estimate of the population mean paper length.

6. An operation manager for a large newspaper wants to determine the proportion of newspapers printed that have a nonconforming attribute, such as excessive rub-off, improper page setup, missing pages, and duplicate pages. The operation manager determines that a random sample of 200 newspapers should be selected for analysis. Suppose that, of this sample of 200, 35 contain some type of non conformance. If the operations manager wants to have 90% confidence of estimating the true population proportion. Set up the interval estimate.

Critical values of Z Level of significance(α) 10% 5% 1%

Critical values for two-tailed test

±1.645 ±1.96 ±2.58

Critical values for left-tailed test

-1.28 -1.645 -2.33

Critical values for right-tailed test

1.28 1.645 2.33

Test of hypothesis

Hypothesis Statements about characteristics of populations, denoted as H.Types of Hypothesis Null & Alternative hypothesis Simple & Composite hypothesis

Hypothesis TestingNull Hypothesis-

The hypothesis actually tested is called the null hypothesis. It is denoted as H0. It is the claim that is initially assumed to be true.

Alternative Hypothesis-

The other hypothesis, assumed true if the null is false, is the alternative hypothesis. It is denoted as H1 or Ha . Ha may usually be considered the researcher’s hypothesis. These two hypotheses are mutually exclusive and exhaustive so that one is true to the exclusion of the other.

Possible conclusions from hypothesis-testing analysis are reject H0 or fail to reject H0.

Hypothesis Testing

Simple Hypothesis - It specifies the distribution completely (One tail test)

H0: μ1 = μ2

H1: μ1 > or < μ2

Composite hypothesis-It does not specifies the distribution completely (Two tail test)

H0: μ1 = μ2

H1: μ1 ≠ μ2

Examples of Hypothesis :

Students attendance in the class has an impact on their performance. high-income earners usually saves moreYouths are brand conscious.

Rules for HypothesesH0 is always stated as an equality claim involving parameters.

H1 is an inequality claim that contradicts H0.

It may be one-sided (using either > or <) or two-sided (using ≠).

A test of hypotheses is a method for using sample data to decide whether the null hypothesis should be rejected.

Rejection region - Values of the test statistic for which we reject the null in favor of the alternative hypothesis

Errors in Hypothesis Testing

A type I error consists of rejecting the null hypothesis H0 when it was true.

A type II error consists of not rejecting H0 when H0 is false.

ErrorIITypeErrorIType

testtheofPowerlevelconfidence 11

Level α Test

Sometimes, the experimenter will fix the value of also known as the significance level.

A test corresponding to the significance level is called a level α test. A test with significance level α is one for which the type I error probability is controlled at the specified level.

Steps in Hypothesis-Testing Analysis1. State the null hypothesis(H0)2. State the alternative hypothesis (H1 )3. Choose the level of significance4. Choose the sample size5. Choose the appropriate test statistic6. Set up the critical value of test statistic7. Collect the data & calculate the value of test statistic8. Compare calculated value of test statistic with tabulated value of test

statistic whether it falls in acceptance region or rejection region9. Make a decision (either accept or reject the null hypothesis)10. Express the statistical decision in the context of the problem

Large sample test(Z-test)

n

XZ

Single Mean

Difference Mean

Proportion

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Questions for discussion

Q1. A random sample of size 20 is drawn from a normal population with mean 28 and variance 25 has a sample mean 30. Test at 5% level of significance.

Q2. A cable TV operator claims that 45 % of the homes in a city have opted for his services. Before sponsoring advertisements on the local cable channel, a company conducted a survey & found that 200 out of 550 persons were found to have cable TV services from the operator. Test the claim at 10% level of significance?

Q3. A survey has conducted between two places on the hourly wages of laborers. Results of the survey are as follows.Places Mean Hourly Wages S.D Sample

1 Rs.18.95 Rs.3.4 200 2 Rs.19.10 Rs.2.6 175

Test the hypothesis at the 0.05 significance level that there is no difference between hourly wages for the landless laborers in the two places.

nSxt

Single Mean Difference Mean

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Small sample test(t-test)

Small sample test(t-test) Test for single mean

The average breaking strength of steel rods is specified to be 18.5 thousand kg. For this a sample of 14 rods was tested . The mean & standard deviation obtained were 17.85 and 1.955 respectively. Test at 5% level of the significance of the deviation.

Test for difference mean

The average life of sample of 10 electric light bulbs was found to be 1456 hours with standard deviation of 423 hours. A second sample of 17 bulbs chosen from a different batch showed a mean life of 1280 hours with standard deviation of 398 hours. Is there a significant difference between the means of two batches. Test at 5% level of the significance.

Chi-square test

• Chi-square analysis is primarily used to deal with categorical (frequency) data

• We measure the “goodness of fit” between our observed outcome and the expected outcome for some variable

• With two variables, we test in particular whether they are independent of one another using the same basic approach.

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