EQT 272 PROBABILITY AND STATISTICS ROHANA BINTI ABDUL HAMID INSTITUT E FOR ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS Free Powerpoint Templates

STATISTICAL INFERENCES CHAPTER 4 STATISTICAL INFERENCES

4.1 INTRODUCTION The field of statistical inference consist of those methods used to make decisions or to draw conclusions about a population. These methods utilize the information contained in a sample from the population in drawing conclusions. Statistical Inference may be divided into two major areas: parameter estimation and hypothesis testing.

Definition 4.1: An Interval Estimate 4.2 PARAMETER ESTIMATION 4.2 Estimation Definition 4.1: An Interval Estimate In interval estimation, an interval is constructed around the point estimate and it is stated that this interval is likely to contain the corresponding population parameter.

Definition 4.2: Confidence Level and Confidence Interval Each interval is constructed with regard to a given confidence level and is called a confidence interval. The confidence level associated with a confidence interval states how much confidence we have that this interval contains the true population parameter. The confidence level is denoted by

4.2.1 Confidence Interval Estimates for Population Mean

EXAMPLE 4.1

SOLUTION

EXAMPLE 4.2 A publishing company has just published a new textbook. Before the company decides the price at which to sell this textbook, it wants to know the average price of all such textbooks in the market. The research department at the company took a sample of 36 comparable textbooks and collected the information on their prices. this information produced a mean price RM 70.50 for this sample. It is known that the standard deviation of the prices of all such textbooks is RM4.50. Construct a 90% confidence interval for the mean price of all such college textbooks.

SOLUTION

EXAMPLE 4.3 Example 4.3 : The scientist wondered whether there was a difference in the average daily intakes of dairy products between men and women. He took a sample of n =50 adult women and recorded their daily intakes of dairy products in grams per day. He did the same for adult men. A summary of his sample results is listed below. Construct a 95% confidence interval for the difference in the average daily intakes of daily products for men and women. Can you conclude that there is a difference in the average daily intakes of daily products for men and women? Men Women Sample size 50 Sample mean 780 grams per day 762 grams per day Sample standard deviation 35 30

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4.2.3 Confidence Interval Estimates for Population Proportion

EXAMPLE 4.4 Example 4.4 According to the analysis of Women Magazine in June 2005, “Stress has become a common part of everyday life among working women in Malaysia. The demands of work, family and home place an increasing burden on average Malaysian women”. According to this poll, 40% of working women included in the survey indicated that they had a little amount of time to relax. The poll was based on a randomly selected of 1502 working women aged 30 and above. Construct a 95% confidence interval for the corresponding population proportion.

Solution

EXAMPLE 4.5 Example 4.5: A researcher wanted to estimate the difference between the percentages of users of two toothpastes who will never switch to another toothpaste. In a sample of 500 users of Toothpaste A taken by this researcher, 100 said that the will never switch to another toothpaste. In another sample of 400 users of Toothpaste B taken by the same researcher, 68 said that they will never switch to another toothpaste. Construct a 97% confidence interval for the difference between the proportions of all users of the two toothpastes who will never switch.

SOLUTION Toothpaste A : n1 = 500 and x1 = 100 Toothpaste B : n2 = 400 and x2 = 68 The sample proportions are calculated; Thus, with 97% confidence we can state that the difference between the two population proportions is between -0.026 and 0.086.

4.2.5 Error of Estimation and Determining the Sample size Definition 4.3:

EXAMPLE 4.6 Example 4.6: A team of efficiency experts intends to use the mean of a random sample of size n=150 to estimate the average mechanical aptitude of assembly-line workers in a large industry (as measured by a certain standardized test). If, based on experience, the efficiency experts can assume that for such data, what can they assert with probability 0.99 about the maximum error of their estimate?

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Definition 4.4:

EXAMPLE 4.7 Example 4.7: A study is made to determine the proportion of voters in a sizable community who favor the construction of a nuclear power plant. If 140 of 400 voters selected at random favor the project and we use as an estimate of the actual proportion of all voters in the community who favor the project, what can we say with 99% confidence about the maximum error?

Solution

EXAMPLE 4.8 How large a sample required if we want to be 95% confident that the error in using to estimate p is less than 0.05? If , find the required sample size.

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4.3 HYPOTHESIS TESTING 4.3 Hypothesis Testing Hypothesis and Test Procedures A statistical test of hypothesis consist of : 1. The Null hypothesis, 2. The Alternative hypothesis, 3. The test statistic and its p-value 4. The rejection region 5. The conclusion

Definition 4.5: Hypothesis testing can be used to determine whether a statement about the value of a population parameter should or should not be rejected. Null hypothesis, H0 : A null hypothesis is a claim (or statement) about a population parameter that is assumed to be true. (the null hypothesis is either rejected or fails to be rejected.) Alternative hypothesis, H1 : An alternative hypothesis is a claim about a population parameter that will be true if the null hypothesis is false. Test Statistic is a function of the sample data on which the decision is to be based. p-value is the probability calculated using the test statistic. The smaller the p-value, the more contradictory is the data to .

Definition 4.6: p-value The p-value is the smallest significance level at which the null hypothesis is rejected.

Developing Null and Alternative Hypothesis It is not always obvious how the null and alternative hypothesis should be formulated. When formulating the null and alternative hypothesis, the nature or purpose of the test must also be taken into account. We will examine: The claim or assertion leading to the test. The null hypothesis to be evaluated. The alternative hypothesis. Whether the test will be two-tail or one-tail. A visual representation of the test itself. In some cases it is easier to identify the alternative hypothesis first. In other cases the null is easier.

Alternative Hypothesis as a Research Hypothesis Many applications of hypothesis testing involve an attempt to gather evidence in support of a research hypothesis. In such cases, it is often best to begin with the alternative hypothesis and make it the conclusion that the researcher hopes to support. The conclusion that the research hypothesis is true is made if the sample data provide sufficient evidence to show that the null hypothesis can be rejected.

Example: A new drug is developed with the goal of lowering blood pressure more than the existing drug. Alternative Hypothesis: The new drug lowers blood pressure more than the existing drug. Null Hypothesis: The new drug does not lower blood pressure more than the existing drug.

Null Hypothesis as an Assumption to be Challenged We might begin with a belief or assumption that a statement about the value of a population parameter is true. We then using a hypothesis test to challenge the assumption and determine if there is statistical evidence to conclude that the assumption is incorrect. In these situations, it is helpful to develop the null hypothesis first.

Example: The label on a soft drink bottle states that it contains at least 67.6 fluid ounces. Null Hypothesis: The label is correct. µ > 67.6 ounces. Alternative Hypothesis: The label is incorrect. µ < 67.6 ounces.

Example: Average tire life is 35000 miles. Null Hypothesis: µ = 35000 miles Alternative Hypothesis: µ  35000 miles

How to decide whether to reject or accept ? The entire set of values that the test statistic may assume is divided into two regions. One set, consisting of values that support the and lead to reject , is called the rejection region. The other, consisting of values that support the is called the acceptance region. Tails of a Test Two-Tailed Test Left-Tailed Test Right-Tailed Test Sign in = < > Rejection Region In both tail In the left tail In the right tail

Steps to perform a test of Hypothesis with predetermined State the null and alternative hypothesis Select the distribution to use Determine the rejection and non-rejection regions Calculate the value of the test statistic Make a decision

4.3.1 a) Testing Hypothesis on the Population Mean, Null Hypothesis : Test Statistic : any population, is known and n is large or normal population, is known and n is small any population, is unknown and n is large normal population, is unknown and n is small

Alternative hypothesis Rejection Region

EXAMPLE 4.9

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b) Hypothesis Test For the Difference between Two Populations Means, Test statistics:

Alternative hypothesis Rejection Region

EXAMPLE 4.10

SOLUTION

4.3.2 a) Testing Hypothesis on the Population Proportion, p Alternative hypothesis Rejection Region

EXAMPLE 4.11 When working properly, a machine that is used to make chips for calculators does not produce more than 4% defective chips. Whenever the machine produces more than 4% defective chips it needs an adjustment. To check if the machine is working properly, the quality control department at the company often takes sample of chips and inspects them to determine if the chips are good or defective. One such random sample of 200 chips taken recently from the production line contained 14 defective chips. Test at the 5% significance level whether or not the machine needs an adjustment.

SOLUTION

b) Hypothesis Test For the Difference between Two Population Proportion,

Alternative hypothesis Rejection Region

EXAMPLE 4.12 Reconsider Example 4.5, At the significance level 1%, can we conclude that the proportion of users of Toothpaste A who will never switch to another toothpaste is higher than the proportion of users of Toothpaste B who will never switch to another toothpaste?

SOLUTION