Business statistics

Statistical Inference: Hypothesis Testing

Lecture Outline

Ø Understand hypothesis-testing procedure using one-tailed and two- tailed tests

Ø Understand the concepts of Type I and Type II errors in hypothesis testing

Ø Understand the procedure of hypothesis testing

The Concept of Normal Distribution

Introduction to Hypothesis Testing

Ø A statistical hypothesis is an assumption about an unknown population parameter.

Ø Hypothesis testing is a well defined procedure which helps us to decide objectively whether to accept or reject the hypothesis based on the information available from the sample.

Ø In statistical analysis, we use the concept of probability to specify a probability level at which a researcher concludes that the observed difference between the sample statistic and the population parameter is not due to chance.

Hypothesis Testing Procedure Seven steps of hypothesis testing

Step 1: Set Null and Alternative Hypotheses

Ø The null hypothesis generally referred by H0 (H sub-zero), is the hypothesis which is tested for possible rejection under the assumption that is true. Theoretically, a null hypothesis is set as no difference or status quo and considered true, until and unless it is proved wrong by the collected sample data.

Ø Symbolically, a null hypothesis is represented as:

Ø The alternative hypothesis, generally referred by H1 (H sub-one), is a logical opposite of the null hypothesis.

Ø Symbolically, alternative hypothesis is represented as:

Step 2: Determine the Appropriate Statistical Test

Ø Type, number, and the level of data may provide a platform for deciding the statistical test.

Step 3: Set the Level of Significance

Ø The level of significance generally denoted by α is the probability, which is attached to a null hypothesis, which may be rejected even when it is true.

Ø The level of significance is also known as the size of the rejection region or the size of the critical region.

Ø The levels of significance which are generally applied by researchers are: 0.01; 0.05; 0.10.

Type I and Type II Errors

When a researcher tests statistical hypotheses, there can be four possible outcomes as follows:

Step 4: Set the Decision Rule

Critical region is the area under the normal curve, divided into two mutually exclusive regions. These regions are termed as acceptance region (when the null hypothesis is accepted) and the rejection region or critical region (when the null hypothesis is rejected).

Acceptance and rejection regions of null hypothesis (two-tailed test)

Two-Tailed Test of Hypothesis

Ø Let us consider the null and alternative hypotheses as below:

Ø Two-tailed tests contain the rejection region on both the tails of the sampling distribution of a test statistic. This means a researcher will reject the null hypothesis if the computed sample statistic is significantly higher than or lower than the hypothesized population parameter (considering both the tails, right as well as left).

Acceptance and rejection regions (alpha = 0.05)

One-Tailed Test of Hypothesis

Let us consider a null and alternative hypotheses as below:

One-tailed test contains the rejection region on one tail of the sampling distribution of a test statistic. In case of a left-tailed test, a researcher rejects the null hypothesis if the computed sample statistic is significantly lower than the hypothesized population parameter.

In case of a right-tailed test, a researcher rejects the null hypothesis if the computed sample statistic is significantly higher than the hypothesized population parameter.

Acceptance and rejection regions for one-tailed (left) test (alpha = 0.05)

Acceptance and rejection regions for one-tailed (right) test (alpha = 0.05)

Step 5: Collect the Sample Data

Ø In this stage of sampling, data are collected and the appropriate sample statistics are computed.

Ø The first four steps should be completed before collecting the data for the study.

Ø It is not advisable to collect the data first and then decide on the stages of hypothesis testing.

Step 6: Analyse the data

Ø In this step, the researcher has to compute the test statistic. This involves selection of an appropriate probability distribution for a particular test.

Ø Some of the commonly used testing procedures are z, t, F, and χ2.

Step 7: Arrive at a Statistical Conclusion and Business Implication

Ø In this step, the researchers draw a statistical conclusion. A statistical conclusion is a decision to accept or reject a null hypothesis.

Ø Statisticians present the information obtained using hypothesis- testing procedure to the decision makers. Decisions are made on the basis of this information. Ultimately, a decision maker decides that a statistically significant result is a substantive result and needs to be implemented for meeting the organization’s goals.

Hypothesis Testing for a Single Population Mean Using the Z Statistic

Ø When sample size is greater than (equals to) 30.

Ø Population has a normal distribution.

Hypothesis Testing for a Single Population Mean Using the Z Statistic

A marketing research firm conducted a survey 10 years ago and found that the average household income of a particular geographic region is Rs 10,000. Mr.Ahmad, who has recently joined the firm as a vice president has expressed doubts about the accuracy of the data. For verifying the data, the firm has decided to take a random sample of 200 households that yield a sample mean (for household income) of Rs 11,000. Assume that the population standard deviation of the household income is Rs 1200. Verify Mr. Ahmad’s doubts using the seven steps of hypothesis testing. Let α = 0.05 (5%).

Example (Solution)

Hypothesis Testing for a Single Population Mean Using the T Statistic (Case of a Small Random

Sample When N < 30)

When a researcher draw a small random sample (n < 30) to estimate the population mean μ and when the population standard deviation is unknown and population is normally distributed, t-test can be applied.

Royal Tyres has launched a new brand of tyres for tractors and claims that under normal circumstances the average life of the tyres is 40,000 km. A retailer wants to test this claim and has taken a random sample of 8 tyres and the mean was found as 39,750 (S.D = 2618.61). He tests the life of the tyres under normal circumstance. The results obtained are presented in Table 10.4.

Example

Example (Solution)

Figure : Computed and critical t values for Example 10.4

A cable TV network company wants to provide modern facilities to

its consumers. The company has five-year old data which reveals

that the average household income is SR 120,000. Company

officials believe that due to the fast development in the region, the

average household income might have increased. The company

takes a random sample of 16 households to verify this assumption.

From the sample the average income of the households is

calculated as 125,000. From historical data, standard deviation is

obtained as 1200. Use alpha = 5% to verify the finding.

Lets Do It !!

Statistical Inference: Hypothesis Testing for

Two Populations

Hypotheseis Testing for the Difference Between Two Population Means Using the Z Statistic (Case of a large Random Sample, n1, n2 > 30, When Population Standard Deviation Is Known)

When sample size is large (n1, n2 > 30) and samples are independent (not related) and the population standard deviation is known, the Z statistic can be used to test the hypothesis for difference between two population means.

The amount of a certain trace element in blood is known to vary with a standard deviation of 14.1 ppm (parts per million) for male blood donors and 9.5 ppm for female donors. Random samples of 75 male and 50 female donors yield concentration means of 28 and 33 ppm, respectively. What is the likelihood that the population means of concentrations of the element are the same for men and women?

LET’S DO IT !

Dominos wanted to test their claim regarding who can eat more slices of Pizza in a Pizza eating festival for males vs females. For the purpose, they randomly selected 22 males and 20 females. The average number of slices eaten by males were 450 with a standard deviation of 25 (from historical data) and the average number of slices eaten by females were 550 with a standard deviation of 20. On the basis of the samples taken for the study, estimate the difference in population means taking 5% as the level of significance and help Dominos to check their claim that females eat more pizza slices than males.

Your Turn !

Hypotheseis Testing for the Difference Between Two Population Means Using the t Statistic (Case of a Small Random Sample, n1, n2 < 30, When Population Standard Deviation Is Unknown)

When sample size is small (n1, n2 < 30) and samples are independent (not related) and the population standard deviation is unknown, the t statistic can be used to test the hypothesis for difference between two population means.

Varun Constructions is a leading company in the construction sector in Afghanistan. It wants to construct flats in USA and Germany and believe that US people have more willingness to pay the money. They formed a team which randomly selected 15 potential customers from USA and 17 customers from Germany and posed the question, “how much dollars are you willing to spend on a flat?” The mean for USA was 1120 (s.d = 350) and for Germany it was found as 1005 (s.d = 255). On the basis of the samples taken for the study, estimate the difference in population means taking 95% as the confidence level.

LET’S DO IT !

Statistical Inference About the Difference Between the Means of Two Related Populations

(Matched Samples)

Ø For dependent samples or related samples, it is important that the two samples taken in the study are of the same size.

Statistical Inference About the Difference Between the Means of Two Related Populations

(Matched Samples)

Ø t Formula to test the difference between the means of two related populations (matched samples) with df = n-1

Being a Marketer for Prozac, suppose you wish to test the effect of Prozac on the well-being of depressed individuals, using a standardised "well-being scale" that sums Likert - type items to obtain a score that could range from 0 to 20. Higher scores indicate greater well- being (that is, Prozac is having a positive effect). Assume the sample population deviation difference to be 2.45 for both the samples. Use level of significance as 5%.

Example

An electronic goods company arranged a special training programme for one segment of its employees. The company wants to measure the change in the attitude of its employees after the training. For this purpose, it has used a well- designed questionnaire, which consists of 10 questions on a 1 to 5 rating scale (1 is strongly disagree and 5 is strongly agree). The company selected a random sample of 10 employees. The scores obtained by these employees are given in Table with S.D of 4.44.

Use α = 0.10 to determine whether there is a significant change in the attitude of employees after the training programme.

Example

A pharmaceutical company wants to diversify into hospitality industry. The company has a notion that the average daily hotel room rates are different in Delhi and Mumbai. The company has taken a random sample of 15 hotels from Delhi and 17 hotels from Mumbai for testing its notion. The average hotel room rates for Delhi is 1572 with a sample variance of 8735 while the average hotel room rates for Mumbai is 1175 with a sample variance of 6126. Take alpha = 10% and test whether there is a difference in the average daily hotel room rates of the two cities taken for the study using seven stages.

PUZZLE 1

The best selling product of a consumer durables manufacturer has reached the saturation stage in its product life cycle. The company is not willing to withdraw the product from the market and has decided to motivate its sales executives to take the personal selling route. The company organized a one month workshop to motivate its sales executive. Three month later, the company selected 18 sales executives and collected data on the number of average productive sales calls in a day before and after the training. Assume the value of average difference for all the sales persons as -3.58 with a standard deviation of 1.64. Test the hypothesis whether the Training workshop has an impact on sales or not?

PUZZLE 2

THANK YOU!