Nihar Ranjan Roy

Nihar Ranjan Roy

 Business Statistics, By Ken Black, Wiley India Edition

Text Book

 Very often when a business researcher is gathering data to test hypotheses about a single population mean, the value of the population standard

deviation is unknown and the researcher must use the sample standard deviation as an estimate of it.

 In such cases, the z test cannot be used.

 t distribution, can be used to analyze hypotheses about a single population mean when σ is unknown if the population is normally distributed for the measurement being studied.

 In this section, we will examine the t test for a single population mean.

TESTING HYPOTHESES ABOUT A POPULATION MEAN USING THE t STATISTIC

( σ UNKNOWN)

 The formula for t test is as below

TESTING HYPOTHESES ABOUT A POPULATION MEAN USING THE t STATISTIC

( σ UNKNOWN)

The U.S. Farmers’ Production Company builds large harvesters. For a harvester to be properly balanced when operating, a 25-pound plate is installed on its side. The machine that produces these plates is set to yield plates that average 25 pounds. The

distribution of plates produced from the machine is normal. However, the shop supervisor is worried that the machine is out of adjustment and is

producing plates that do not average 25 pounds.

To test this concern, he randomly selects 20 of the plates produced the day before and weighs them.

Table on left shows the weights obtained, along with the computed sample mean and sample standard deviation.

Problem

The test is to determine whether the machine is out of control,

and the shop supervisor has not specified whether he believes the machine is producing plates that are too heavy or too light. Thus a two-tailed test is appropriate. The following hypotheses are

tested.

H

: µ= 25 pounds H

: µ≠ 25 pounds An α of .05 is used.

Solution

 Because n = 20, the

degrees of freedom for this test are 19 (20 - 1).

 The t distribution table is a one-tailed table but the test for this problem is two tailed, so alpha must be split, which yields α/2

= .025, the value in each tail. (To obtain the table t value when conducting a two-tailed test, always split alpha and use 2.)

 The table t value for this example is 2.093.

t.

= 2.093

Pictorial representation of rejection region

 In this case, the decision rule is to reject the null hypothesis if the observed value of t is less than -2.093 or greater than +2.093 (in the tails of the

distribution).

 Because the observed t value is +1.04, the null hypothesis is not rejected. Not enough evidence is found in this sample to reject the hypothesis that the

population mean is 25 pounds.

Calculate t

Figures released by the U.S. Department of Agriculture show that the average size of farms has increased since 1940. In 1940, the mean size of a farm was 174 acres; by 1997, the average size was 471 acres. Between those years, the number of farms

decreased but the amount of tillable land remained relatively constant, so now farms are bigger. This trend might be explained, in part, by the inability of small farms to compete with the prices and costs of large-scale operations and to produce a level of income necessary to support the farmers’ desired standard of living. Suppose an

agribusiness researcher believes the average size of farms has now increased from the 1997 mean figure of 471 acres. To test this notion, she randomly sampled 23 farms across the United States and ascertained the size of each farm from county records.

The data she gathered follow. Use a 5% level of significance to test her hypothesis.

Assume that number of acres per farm is normally distributed in the population.

 445 489 474 505 553 477 454 463 466

 557 502 449 438 500 466 477 557 433

 545 511 590 561 560

Problem

Solution

 Based on population figures and other general information on the U.S.

population, suppose it has been estimated that, on average, a family of four in the United States spends about $1135 annually on dental expenditures.

Suppose further that a regional dental association wants to test to determine if this figure is accurate for their area of the country. To test this, 22 families of four are randomly selected from the population in that area of the country and a log is kept of the family’s dental expenditures for one year. The resulting

data are given below. Assuming that dental expenditures are normally distributed in the population, use the data and an alpha of .05 to test the dental association’s hypothesis.

1008 812 1117 1323 1308 1415

831 1021 1287 851 930 730

699 872 913 944 954 987

1695 995 1003 994

Problem

 n = 22 α= .05 ̅ = 1031.32 s = 240.37 df = 22 – 1 = 21

Solution

 Data analysis used in business decision making often contains proportions to

describe such aspects as market share, consumer makeup, quality defects, on-time delivery rate, profitable stocks, and others.

 Business surveys often produce information expressed in proportion form, such as .45 of all businesses offer flexible hours to employees or .88 of all businesses have Web sites.

 Other examples of hypothesis testing about a single population proportion might include:

−A market researcher wants to test to determine whether the proportion of new car purchasers who are female has increased.

−A financial researcher wants to test to determine whether the proportion of companies that were profitable last year in the average investment officer’s portfolio is .60.

−A quality manager for a large manufacturing firm wants to test to determine whether the proportion of defective items in a batch is less than .04.

Hypothesis testing about proportion

 ̅ denotes a sample proportion and

 p denotes the population proportion.

 To validly use this test, the sample size must be large enough such that

 and n.p ≥5 and n.q ≥ 5.

At times a researcher needs to test hypotheses about a population variance.

For example, in the area of statistical quality control, manufacturers try to produce equipment and parts that are consistent in measurement.

Suppose a company produces industrial wire that is specified to be a particular thickness. Because of the production process, the thickness of the wire will vary slightly from one end to the other and from lot to lot and batch to batch.

Even if the average thickness of the wire as measured from lot to lot is on specification, the variance of the measurements might be too great to be

acceptable. In other words, on the average the wire is the correct thickness, but some portions of the wire might be too thin and others unacceptably thick. By conducting hypothesis tests for the variance of the thickness measurements, the quality control people can monitor for variations in the process

TESTING HYPOTHESES ABOUT A VARIANCE

TESTING HYPOTHESES ABOUT A VARIANCE

As an example, a manufacturing firm has been working diligently to implement a just-in-time inventory system for its production line. The final product

requires the installation of a pneumatic tube at a particular station on the

assembly line. With the just in-time inventory system, the company’s goal is to minimize the number of pneumatic tubes that are piled up at the station

waiting to be installed. Ideally, the tubes would arrive just as the operator needs them. However, because of the supplier and the variables involved in

getting the tubes to the line, most of the time there will be some buildup of tube inventory. The company expects that, on the average, about 20 pneumatic tubes will be at the station. However, the production superintendent does not want the variance of this inventory to be greater than 4. On a given day, the number of pneumatic tubes piled up at the workstation is determined eight different times and the following number of tubes are recorded.

23 17 20 29 21 14 19 24

Problem

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