Testing Significance Of Mean

T-test for one population Mean Dr.Srilakshminarayana.G

An automobile manufacturer substitutes a different engine in cars that were known to have an average miles-per-gallon rating of 31.5 on the highway. The manufacturer wants to test whether the new engine changes the miles-per-gallon rating of the automobile model. A random sample of 100 trial runs gives sample mean= 29.8 miles per gallon and s=6.6 miles per gallon. Using the 0.05 level of significance, is the average miles-per-gallon rating on the highway for cars using the

new engine different from the rating for cars using the old engine?

A certain commodity is known to have a price that is stable through time and does not change according to any known trend. Price, however, does change from day to day in a random fashion. If the price is at a certain

level one day, it is as likely to be at any level the next day within some probability bounds approximately given by a normal distribution. The mean daily price is believed to be $14.25. To test the hypothesis that the

average price is $14.25 versus the alternative hypothesis that it is more than $14.25, a random sample of 16 daily prices is collected. The results are sample mean= $16.50 and s=$4.2. Using level of significance 0.05, can

you reject the null hypothesis?

Many recent changes have affected the real estate market. A study was undertaken to determine customer satisfaction from real estate deals.

Suppose that before the changes, the average customer satisfaction rating, on a scale of 0 to 100, was 77. A survey questionnaire was sent to a random sample of 50 residents who bought new plots after the changes in the market were instituted,

and the average satisfaction rating for this sample was found to be 84; the sample standard deviation was found to be s=28. Use a level of significance of your choice, and determine whether statistical evidence indicates a change in customer

satisfaction. If you determine that a change did occur, state whether you believe customer satisfaction has improved.

A study was undertaken to evaluate how stocks are affected by being listed in the Standard & Poor’s 500 Index. The aim of the study was to assess average excess returns for these stocks, above returns on the market as a whole. The average excess return on any stock is zero because the “average” stock moves with the market as a whole. As part of the study, a random sample of 13 stocks newly included in the S&P 500 Index was selected. Before the sampling takes place, we allow that average “excess return” for stocks newly listed in the Standard & Poor’s 500 Index may be either positive or negative; therefore, we want to test the null hypothesis that average excess return is equal to zero versus the alternative that it is not zero. If the excess return on the sample of 13 stocks averaged 3.1% and had a standard deviation of 1%, do you believe that inclusion in the Standard & Poor’s 500 Index changes a stock’s excess return on investment, and if so, in which direction? Explain. Use level of significance 0.05.

In controlling the quality of a new drug, a dose of medicine is supposed to contain an average of 247 parts per million (ppm) of a certain chemical. If the concentration is higher than 247 ppm, the drug may cause some side effects; and if the concentration is below 247 ppm, the drug may be ineffective. The manufacturer wants to check whether the average concentration in a large shipment is the required 247 ppm or not. A random sample of 60 portions is tested, and the sample mean is found to be 250 ppm and the sample standard deviation 12 ppm. Test the null hypothesis that the average concentration in the entire large shipment is 247 ppm

versus the alternative hypothesis that it is not 247 ppm using a level of significance 0.05. Do the same using level of significance 0.01. What is your conclusion? What is your decision about the shipment? If the shipment were guaranteed to contain an average concentration of 247 ppm, what would your decision be, based on the statistical hypothesis test? Explain.

• The Boston Transit Authority wants to determine whether there is any need for changes in the frequency of service over certain bus routes. The transit authority needs to know whether the frequency of service should increase, decrease, or remain the same. The transit authority determined that if the average number of miles travelled by bus over the routes in question by all residents of a given area is about 5 per day, then no change will be necessary. If the average number of miles travelled per person per day is either more than 5 or less than 5, then changes in service may be necessary. The authority wants, therefore, to test the null

hypothesis that the average number of miles travelled per person per day is 5.0 versus the alternative hypothesis that the average is not 5.0 miles. The required level of significance for this test is 0.05. A random sample of 120 residents of the area is taken, and the sample mean is found to be 2.3 miles per resident per day and the sample standard deviation 1.5 miles. Advise the authority on what should be done. Explain your recommendation. Could you state the same result at different levels of significance? Explain.

According to Money, the average appreciation, in percent, for stocks has been 4.3% for the five-year period ending in May 2007. An analyst tests this claim by looking

at a random sample of 50 stocks and finds a sample mean of 3.8% and a sample standard deviation of 1.1%. Using level of significance 0.05, does the analyst have statistical evidence to reject the claim made by the magazine?

Average total daily sales at a small food store are known to be $452.80. The store’s

management recently implemented some changes in displays of goods, order within aisles, and other changes, and it now wants to know whether average sales volume has changed. A random sample of 12 days shows sample mean= $501.90 and s=$65.00. Using level of

significance 0.05, is the sampling result significant? Explain.

New software companies that create programs for World Wide Web applications believe that

average staff age at these companies is 27. To test this two-tailed hypothesis, a random sample is collected: 41, 18, 25, 36, 26, 35, 24, 30, 28, 19, 22, 22, 26, 23, 24, 31, 22, 22, 23, 26, 27, 26, 29, 28, 23, 19, 18, 18, 24, 24, 24, 25, 24, 23, 20, 21, 21, 21, 21, 32, 23, 21, 20 Test, using level of significance 0.05

A new chemical process is introduced by Duracell in the production of lithium-ion batteries. For batteries produced by the old process, the average life of a battery is 102.5 hours. To determine whether the new process affects the average life of the batteries, the manufacturer collects a random sample of 25 batteries produced by

the new process and uses them until they run out. The sample mean life is found to be 107 hours, and the sample standard deviation is found to be 10 hours. Are these results significant at the level of significance 0.05 level? Are they significant at the

level of significance 0.01 level? Explain. Draw your conclusion.

Average soap consumption in a certain country is believed to be 2.5 bars per person per month. The standard deviation of the population is known to be 0.8. While the standard deviation is not believed to have changed (and this may be substantiated by several studies), the mean consumption may have changed either upward or downward. A survey is therefore undertaken to test the null hypothesis that average soap consumption is still 2.5 bars per person per month versus the alternative that it is not. A sample of size n=20 is collected and gives sample mean 2.3. The population is assumed to be normally distributed. What is the appropriate test statistic in this case? Conduct the test and state your conclusion. Use level of significance 0.05. Does the choice of level of significance change your conclusion? Explain.

According to Money, which not only looked at stocks but also compared them with real estate, the average appreciation for all real estate sold in the five years ending May 2007 was 12.4% per year. To test this claim, an analyst looks at a random sample of 100 real estate deals in the period in question and finds a sample mean of 14.1% and a sample standard deviation of 2.6%. Conduct a two tailed test using the 0.05 level of significance. Suppose that the Goodyear Tire Company has historically held 42% of the market for automobile tires in the United States. Recent changes in company operations, especially its diversification to other areas of business, as well as changes in competing firms’ operations, prompt the firm to test the validity of the assumption that it still controls 42% of the market. A random sample of 550 automobiles on the road shows that 219 of them have Goodyear tires. Conduct the test at level of significance 0.01.

The manufacturer of electronic components needs to inform its buyers of the proportion of defective components in its shipments. The company has been stating that the percentage of defectives is 12%. The company wants to test whether the proportion of all components that are defective is as claimed. A random sample of 100 items indicates 17 defectives. Use level of significance 0.05 to test the hypothesis that the percentage of defective components is 12%. According to Business Week, the average market value of a biotech company is less than $250 million. Suppose that this indeed is the alternative hypothesis you want to prove. A sample of 30 firms reveals an average of $235 million and a standard deviation of $85 million. Conduct the test at level of significance 0.05 and level of significance 0.01. State your conclusions.

A company’s market share is very sensitive to both its level of advertising and the levels of its competitors’ advertising. A firm known to have a 56% market share wants to test whether this value is still valid in view of recent advertising campaigns of its competitors and its own increased level of advertising. A random sample of 500 consumers reveals that 298 use the company’s product. Is there evidence to conclude that the company’s market share is no longer 56%, at the 0.01 level of significance? According to a financial planner, individuals should in theory save 7% to 10% of their income over their working life, if they desire a reasonably comfortable retirement. An agency wants to test whether this actually happens with people in the United States, suspecting the overall savings rate may be lower than this range. A random sample of 41 individuals revealed the following savings rates per year: 4, 0, 1.5, 6, 3.1, 10, 7.2, 1.2, 0, 1.9, 0, 1.0, 0.5, 1.7, 8.5, 0, 0, 0.4, 0, 1.6, 0.9, 10.5, 0, 1.2, 2.8, 0, 2.3, 3.9, 5.6, 3.2, 0, 1, 2.6, 2.2, 0.1, 0.6, 6.1, 0, 0.2, 0, 6.8 Conduct the test and state your conclusions. Use the lower value, 7%, in the null hypothesis. Use level of significance 0.01. Interpret.

The theory of finance allows for the computation of “excess” returns, either above or below the current stock market average. An analyst wants to determine whether stocks in a certain industry group earn either above or below the market average at a certain time period. The null hypothesis is that there are no excess returns, on the average, in the industry in question. “No average excess returns” means that the population excess return for the industry is zero. A random sample of 24 stocks in the industry reveals a sample average excess return of 0.12 and sample standard deviation of 0.2. State the null and alternative hypotheses, and carry out the test at the 0.05 level of significance.

According to Fortune, on February 27, 2007, the average stock in all U.S. exchanges fell by 3.3%. If a random sample of 120 stocks reveals a drop of 2.8% on that day and a standard deviation of 1.7%, are there grounds to reject the magazine’s claim?

According to Money, the average amount of money that a typical person in the United States would need to make him or her feel rich is $1.5 million. A researcher wants to test this claim. A random sample of 100 people in the United States reveals that their mean “amount to feel rich” is $2.3 million and the standard deviation is $0.5 million. Conduct the test. The U.S. Department of Commerce estimates that 17% of all automobiles on the road in the United States at a certain time are made in Japan. An organization that wants to limit imports believes that the proportion of Japanese cars on the road during the period in question is higher than 17% and wants to prove this. A random sample of 2,000 cars is observed, 381 of which are made in Japan. Conduct the hypothesis test at level of significance 0.01, and state whether you believe the reported figure.

Airplane tires are sensitive to the heat produced when the plane taxis along runways. A certain type of airplane tire used by Boeing is guaranteed to perform well at temperatures as high as 125°F. From time to time, Boeing performs quality control checks to determine whether the average maximum temperature for adequate performance is as stated, or whether the average maximum temperature is lower than 125°F, in which case the company must replace all tires. Suppose that a random sample of 100 tires is checked. The average maximum temperature for adequate performance in the sample is found to be 121°F and the sample standard deviation 2°F. Conduct the hypothesis test, and conclude whether the company should take action to replace its tires. An advertisement for Qualcomm appearing in various business publications in fall 2003 said: “The average lunch meeting starts seven minutes late.” A research firm tested this claim to see whether it is true. Using a random sample of 100 business meetings, the researchers found that the average meeting in this sample started 4 minutes late and the standard deviation was 3 minutes. Conduct the test using the 0.05 level of significance.

Certain eggs are stated to have reduced cholesterol content, with an average of only 2.5% cholesterol. A concerned health group wants to test whether the claim is true. The group believes that more cholesterol may be found, on the average, in the eggs. A random sample of 100 eggs reveals a sample average content of 5.2% cholesterol, and a sample standard deviation of 2.8%. Does the health group have cause for action?

An advertisement for the Audi TT model lists the following performance specifications: standing start, 0–50 miles per hour in an average of 5.28 seconds; braking, 60 miles per hour to 0 in 3.10 seconds on the average. An independent testing service hired by a competing manufacturer of highperformance automobiles wants to prove that Audi’s claims are exaggerated. A random sample of 100 trial runs gives the following results: standing start, 0–50 miles per hour in an average of 5.8 seconds and s=1.9 seconds; braking, 60 miles per hour to 0 in an average of 3.21 seconds and s=0.6 second. Carry out the two hypothesis tests, state the p-value of each test, and state your conclusions.

Borg-Warner manufactures hydroelectric miniturbines that generate low-cost, clean electric power from the energy in small rivers and streams. One of the models was known to produce an average of 25.2 kilowatts of electricity. Recently the model’s design was improved, and the company wanted to test whether the model’s average electric output had changed. The company had no reason to suspect, a priori, a change in either direction. A random sample of 115 trial runs produced an average of 26.1 kilowatts and a standard deviation of 3.2 kilowatts. Carry out a statistical hypothesis test, give the p-value, and state your conclusion. Do you believe that the improved model has a different average output?

According to the Wall Street Journal, the average American jockey makes only $25,000 a year. Suppose you try to disprove this claim against a right-tailed alternative and your random sample of 100 U.S. jockeys gives you a sample mean of $45,600 and sample standard deviation of $20,000. What is your p-value? The engine of the Volvo model S70 T-5 is stated to provide 246 horsepower. To test this claim, believing it is too high, a competitor runs the engine n=60 times, randomly chosen, and gets a sample mean of 239 horsepower and standard deviation of 20 horsepower. Conduct the test, using level of significance 0.01.

At Armco’s steel plant in Middletown, Ohio, statistical quality-control methods have been used very successfully in controlling slab width on continuous casting units. The company claims that a large reduction in the steel slab width variance resulted from the use of these methods. Suppose that the variance of steel slab widths is expected to be 156 (squared units). A test is carried out to determine whether the variance is above the required level, with the intention to take corrective action if it is concluded that the variance is greater than 156. A random sample of 25 slabs gives a sample variance of 175. Using level of significance 0.05, should corrective action be taken? According to an article in the New York Times, new Internet dating Web sites use sex to advertise their services. One such site, True.com, reportedly received an average of 3.8 million visitors per month. Suppose that you want to disprove this claim, believing the actual average is lower, and your random sample of 15 months revealed a sample mean of 2.1 million visits and a standard deviation of 1.2 million. Conduct the test using level of significance 0.05. What is the approximate p-value?

According to The New York Times,3-D printers are now becoming a reality. If a manufacturer of the new high-tech printers claims that the new device can print a page in 3 seconds on average, and a random sample of 20 pages shows a sample mean of 4.6 seconds and sample standard deviation of 2.1 seconds, can the manufacturer’s claim be rejected? Explain and provide numerical support for your answer.

The tensile strength of parts made of an alloy is claimed to be at least 1,000 kg/cm2 The population standard deviation is known from past experience to be

10 kg/cm2.It is desired to test the claim at a level of significance of 5% with the probability of type II error, restricted to 8% when the actual strength is only 995 kg/cm2. The engineers are not sure about their decision to limit type-II error as described and want to do a sensitivity analysis of the sample size on actual mean ranging from 994 to 997 kg/cm2 and limits on ranging from 5% to 10%. Prepare a plot of the sensitivity.