Chapter 3 Test of Hypothesis 3 Test of Hypothesis 3.1 Introduction A parameter can be estimated from sample data either by a single number (a point estimate) or an entire interval of plausible values (a confidence interval) as discussed in Chapter 2. In making decisions about the value of a parameter, such as a population mean or a population proportion, inferential statistics are commonly used. One of the most commonly used methods for making such decisions is to perform a hypothesis test. A hypothesis is a belief about a population, or state of nature. The way that statistics are used to form a decision about a population parameter is called the statistical hypothesis test. In any hypothesis-testing problem, there are two contradictory hypotheses under consideration. For example, a claim that = 1000, where is the true mean income in RM for a general worker in Malaysia. The contradictory claim then is > 1000, which means that the income has increased more than what is claimed initially. The objective is to decide, based on sample information, which of the two hypotheses is correct. The problem will be formulated so that one of the claims is initially favored. This initially favored claim will not be rejected in favor of the alternative claim unless sample evidence contradicts it and provides strong support for the alternative assertion. 53 Chapter 3 Test of Hypothesis Before we proceed further, it is important to define the statistical phrases that will be used frequently in this chapter. Hypothesis Test: It is a process of using sample data and statistical procedures to decide whether to reject or not to reject a hypothesis (statement) about a population parameter value (or about its distribution characteristics). Null hypothesis, H0: This is a statement in which a population parameter has a specific value. The null hypothesis is so named because it is the “starting point” for the investigation. The null hypothesis is initially assumed to be true. Therefore, it is the hypothesis to be tested. Alternative Hypothesis, H1: It is a statement about the same population parameter that is used in the null hypothesis and generally this is a statement that specifies the population parameter has a value different in some way, from the value given in the null hypothesis. The rejection of the null hypothesis will imply the acceptance of this alternative hypothesis. Test statistic It is a function of the sample data on which the decision is to be based. Critical region It is a set of values of the test statistics for which the null hypothesis will be rejected. 54 Chapter 3 Test of Hypothesis Critical value (point) It is the “first” (or “boundary”) value in the critical region. Type I and Type II Errors Whenever statistical inference methods are employed, it is always possible that the decision reached will be incorrect. This is because partial information obtained from the sample is used to draw conclusions about the entire population. Hence, there are two types of errors, namely Type I and Type II errors that may have to be considered when performing the hypothesis testing. Ho is True False Do not reject Ho Correct decision Type II error Reject Ho Type I error Correct decision Type I error: Rejecting Ho when it is in fact true. Type II error: Accepting Ho when it is in fact false. The probability of making such errors may be labeled as follows: (significance level ) is the probability of making a Type I error. is the probability of making Type II error. 3.2 Procedures for hypothesis testing 1. Define the question to be tested and formulate a hypothesis for stating the problem. Ho : a H1 : a or a or > a 55 Chapter 3 Test of Hypothesis 2. Choose the appropriate test statistic and calculate the sample statistics value. The choice of test statistics is dependent upon the probability distribution of the random variable involved in the hypothesis. 3. Establish the test criterion by determining the critical value and critical region. 4. Draw conclusions, whether to accept or to reject the null hypothesis. 3.3 One tail test and Two tail test 1. Reject Ho : a Ho H1 : a Reject Ho when Ztest Z Z 2. Ho : a H1 : a Reject Ho Reject Ho when Ztest Z -Z 3. Ho : a H1 : a Reject Ho Reject Ho Reject Ho when | Ztest | Z / 2 -Z / 2 56 Z/2 Chapter 3 Test of Hypothesis 3.4 P-Values in Hypothesis Tests In reporting the results of a hypothesis test we normally state whether the null hypothesis was or was not rejected. However, if for example the null hypothesis was rejected, no information given whether the computed value of the test statistics was just barely in the rejection region or very far from the region. Hence, the P-value approach has been adopted. The P-value is the smallest level of significance that would lead to rejection of the null hypothesis H0 with the given data. Thus the value would convey much information about the weight of evidence against H0 and so a decision maker can draw a conclusion at any specified level of significance. If ztest is the computed value of the test statistics, the P-value is 21 ( z test for a two - tailed test : H 0 : 0 P 1 ( z test for one - tailed test (upper) : H 0 : 0 for one - tailed test (lower) : H 0 : 0 ( z test Φ( z test P(Z z test ) where Z is N (0,1). H1 : 0 H1 : 0 H1 : 0 For example, let say ztest = 2.15 and the alternative hypothesis is a two-tailed test, then the P-value is P value = 2[1- (2.15) ] = 0.0316. The H 0 would be rejected at any level of significance P -value. For example, H 0 would be rejected if = 0.05, but it would not be rejected if =0.01. 57 Chapter 3 Test of Hypothesis 3.5 Connection between Hypothesis Tests and Confidence Interval There is a close relationship between the test of a hypothesis about any parameter, say and the confidence interval for . If [l,u] is a 100(1- )% confidence interval for the parameter , the test size of the hypothesis H 0 : 0 H1 : 0 will lead to rejection of H 0 if and only if 0 is not in the 100(1- )% CI[l,u]. Even though, hypothesis tests and CIs are equivalent procedures insofar as decision making or inference about parameter is concerned, each provides somewhat different insights. The CI provides a range of likely values for parameter at a stated confidence level, whilst the hypothesis testing provides framework for displaying the risks levels such as the P-value associated with a specific decision. Example 3.1 The standard mean weight of a new laptop is 1.8kg. A quality control officer inspected 50 laptop after receiving complaints from customers that the mean weight was less than 1.8k g. Formulate the hypotheses (null and alternatives) and establish the test criterion at 5 % significance level. Solution 58 Chapter 3 Test of Hypothesis Example 3.2 In 1990 the mean retail price of all local magazine was RM 10. This year’s retail prices for 50 randomly selected local magazines have increased over 1990 mean of RM 10. Formulate the hypotheses (null and alternatives) and establish the test criterion at 1% significance level. Solution Example 3.3 A marketing firm in JB has told a potential investor that the mean cost of a twostorey shop house in JB is RM 500,000. To check this claim, the investor randomly selects 30 two-storey shop houses in JB and found that the firm’s claim is incorrect. Formulate the hypotheses (null and alternatives) and establish the test criterion at 5% significant level. Solution 59 Chapter 3 Test of Hypothesis 3.6 Hypothesis Tests Involving Single Population mean µ. The Hypotheses: H 0 : 0 H1 : 0 Reject H0 when Z test Z H1 : 0 Reject H0 when Z test Z H1 : 0 Reject H0 when Z test Z 2 The Test Statistics: To choose the right test statistics for a particular problem, the information on population variance, 2 and sample size, n is necessary. 1. If the population variance, 2 is known, then the test statistic to be used is : z test X 0 n 2. If the population variance, 2 is unknown and the sample size is large, i.e. n 30 , then the test statistics to be used is z test X 0 s n 60 Chapter 3 Test of Hypothesis 3. If the population variance, 2 is unknown and the sample size is small, i.e. n 30 , then the test statistics to be used is t test X 0 s n Example 3.4 The mean amount of time spent on sewing embroideries on a piece of table cloth is 25 hours with a standard deviation of 3 hours. However, according to a tailor, the mean amount of time has decreased due to the more sophisticated machines used in sewing embroideries. A sample of 27 table cloths was selected at random and the mean number of hours spent on sewing embroideries on a table cloth was found to be 24 hours. Can the tailor’s claim be accepted at 5% signifcance level? Solution Example 3.5 It was claimed that the mean cost per meal at the cafeteria was cheaper and was at most RM5. To support the claim, 40 college students who ate in the cafetaria were selected at random and the mean cost per meal was RM 4.50 with a standard deviation of RM 1. Can we accept the claim at 1% significance level? Solution 61 Chapter 3 Test of Hypothesis Example 3.6 Ahmad works in a quality control department. His job involves checking the packaging and each package should weigh 100g. 12 randomly selected packages are taken and their weights in g are recorded as follows: 100 102 100 100 99 95 97 100 95 98 102 97 Perform the required hypothesis test at 5% significance level to check whether the mean weight per package is not equal to 100g. Solution 62 Chapter 3 Test of Hypothesis Example 3.7 The mean water temperature downstream from a power plant cooling tower discharge pipe should be no more than 100°F. Past experience has indicated that the standard deviation is 2°F. The water temperature is measured on nine randomly chosen days, and the average temperature is found to be 98°F. a. Should the water temperature be judged acceptable with α = 0.05? b. What is the P-value for this test? c. What is the probability of accepting the null hypothesis at α = 0.05 if the water has a true mean temperature of 104°F? Solution 3.7 Hypothesis Test Involving Differences of Two Population Means, 1 The Hypotheses: H 0 : 1 2 0 H 1 : 1 2 0 Reject H0 when Z test Z H 1 : 1 2 0 Reject H0 when Z test Z H 1 : 1 1 0 Reject H0 when Z test Z 2 63 2 Chapter 3 Test of Hypothesis The Test Statistics: Hypothesis test involving two population means, 1 2 can be categorized as follows: 1. If the population variances, 12 dan 22 are known and both n1 and n2, are samples of any sizes, then the test statistic to be used is: z test X1 X 2 0 12 n1 2. 22 n2 If the population variances, 12 dan 22 are unknown, then the following table shows the different formulaes that may be used depending on the sample sizes and the assumption on the population variances. Sample sizes Equality of variances n1 30, n2 30 z test X1 X 2 0 t test s12 s 22 n1 n2 12 22 n1 30, n2 30 z test X1 X 2 0 sg 12 22 sg s12 s 22 n1 n2 s12 s 22 n1 n2 2 n1 n2 2 2 X 1 X 2 0 sg n1 1s12 n2 1s 22 sg 1 1 n1 n 2 n1 1s12 n2 1s 22 n1 n2 2 n1 n2 2 64 2 s12 s 22 n1 n2 n1 1 n2 1 t test 1 1 n1 n2 X1 X 2 0 Chapter 3 Test of Hypothesis Example 3.8 The mean lifetime of 30 batteries produced by Company XYZ is 50 hours and the mean lifetime of 35 batteries produced by Company ABC is 48 hours. If the standard deviation of all batteries produced by Company XYZ is 3 hours and the standard deviation of all batteries batteries produced by Company XYZ is better than that of Company ABC. Assume the data was taken from a normal distribution. Solution Example 3.9 A statistics test was given to two classes of 45 and 55 students respectively. In the first class the mean grade was 80 with a standard deviation of 8, while in the second class the mean grade was 85 with a standard deviation of 7. Is there a significant difference between the performances of the two classes at 5% level of significance? Assume the two population variances are equal. Solution 65 Chapter 3 Test of Hypothesis Example 3.10 A sample of 60 maids from country A earns an average of RM 400 per week with a standard deviation of RM 16, while a sample of 60 maids from country B earns an average of RM 300 per week with a standard deviation of RM 15. Test at 5% significance level that country A maids’ average earning exceed country B maids’ average earning more than RM 40 per week. Solution 66 Chapter 3 Test of Hypothesis Example 3.11 An experiment was performed to compare the strength of two different materials. 12 pieces of material A and 10 pieces of material B were tested by exposing each piece to a machine. The samples of material A gave an average strength of 85 units and standard deviation of 4, while the samples of material B gave an average strength of 81 units and standard deviation of 5. Can we conclude at level of significance α=0.05 that the strength of material A exceeds that of material B by more than 2 units? Assume the populations to be approximately normal with equal variances. Solution 67 Chapter 3 Test of Hypothesis Example 3.12 The following data are the marks scored by the participants in one cooking competition organized by a famous instant noodle company Mi Kari. Johor sent 9 groups while Malacca sent 6 groups in that competition. Johor 50 68 44 78 63 46 Malacca 55 70 62 55 65 56 69 54 50 Assuming that the two population variances are not equal, test that the mean scores earned by Johor is not more than the mean scores earned by Malacca at 5% significance level. Assume the data follows a normal distribution. Solution 68 Chapter 3 Test of Hypothesis 3.8 Hypothesis Test for Single Population Proportion π for large samples ( n 30 ). The Hypotheses: H0 : 0 H1 : 0 Reject H0 when Z test Z H1 : 0 Reject H0 when Z test Z H1 : 0 Reject H0 when Z test Z 2 The Test Statistic: ztest p 0 0 1 0 , p is a sample proportion n Example 3.13 A factory that manufactures semiconductor products requires a fraction of defectives at a critical manufacturing step to be less than 0.05. If so, the manufacturer successfully demonstrates process capability. The manufacturer takes a random sample of 200 devices and finds that four of them are defective. Can the manufacturer demonstrate process capability for the customer? Assume samples are taken from a normal distribution. Use α = 0.05. Solution 69 Chapter 3 Test of Hypothesis 3.9 Hypothesis Tests Involving differences of two Population Proportions, π1-π2 for large samples (n 30, n 30) . 1 2 H0 : 1 2 0 H1 : 1 2 0 Reject H0 when Z test Z H1 : 1 2 0 Reject H0 when Z test Z H1 : 1 2 0 Reject H0 when Z test Z 2 The null hypothesis is : H 0 : 1 2 0 (population proportions are equal) If the null hypothesis is true, then 1 2 0 , and so the standardized random variable becomes z p p 1 2 1 1 n 1 where n 2 = p p 1 2 1 1 (1 ) n n 1 2 denotes the common value of 1 and 2 . The random variable above cannot be used as the test statistics since is unknown. Consequently, must be estimated using sample information. The best estimate of is obtained by pooling the data to get the proportion in both samples combined; that is we estimate by pp x1 x2 n1 n2 We call p p the pooled sample proportion. Hence, the test statistics for two populations’ proportion is as follows: 70 Chapter 3 Test of Hypothesis The Test Statistics z test p p 1 2 0 1 1 p p (1 p p ) n n 1 2 Example 3.14 Two different types of machines are used to form car spare parts. A part is considered defective if it has distorted shapes. Two random sample each of size 300 are selected and 15 defective parts are found in the sample from machine A while 9 defective parts are found in the sample from machine B. Is it reasonable to conclude that both machines produce the same fraction of defective parts, using α = 0.05? Assume data are taken from normal distribution. Solution 71 Chapter 3 Test of Hypothesis 3.10 Hypothesis Test Involving Population Variance, 2. Single The Hypotheses: H 0 : 2 02 H 1 : 2 02 Reject 2 2 ,n 1 H 0 if test H 1 : 2 02 Reject 2 12 ,n 1 H 0 if test 2 test 2 2,n 1 H 1 : 2 02 Reject H 0 if or 2 test 12 2,n 1 The Test Statistic: 2 test n 1s 2 02 Example 3.15 A random sample of 10 filled plastic cups were taken and the volume are as follows, in fluid ounces (fl.oz). 12.00 12.01 12.00 11.99 11.89 11.85 12.01 11.99 12.05 11.58 If the variance to fill volume exceeds 0.01 fl.oz, the machine must be reset. a. Is there evidence in the sample data to suggest there has a problem with the filling machine? Use α = 0.05 and assume that the fill volume follows normal distribution. 72 Chapter 3 Test of Hypothesis b. If α = 0.01, will you change the conclusions you have made in (a)? Solution: 3.11 Hypothesis Test Involving the ratio of two Population Variances, . 2 1 2 2 The Hypotheses: H 0 : 12 22 H 1 : 12 22 Reject H 0 if Ftest F ,n1 1,n2 1 H 1 : 12 22 Reject H 0 if Ftest F1 ,n1 1,n2 1 Ftest F 2,n1 1, n2 1 H1 : 2 1 2 2 Reject H 0 with F1 ,n1 1,n2 1 if or Ftest F1 2,n1 1,n2 1 1 F ,n2 1,n1 1 The Test Statistic: Ftest s12 s22 73 Chapter 3 Test of Hypothesis Example 3.16 A machine could cut 12 glass rods before it was serviced. The observed length for the 12 rulers had a variance of 3.6 cm 2. However, after the machine was serviced 10 glass rods were cut and the variance of its length was 2.5 cm2. Did servicing the machine improve its operation? Test at 5% significance level. Solution Example 3.17 A weight watcher trainer has two groups of dieters in his weight-loss program. Group A has 10 dieters while Group B has 20 dieters. After one month the weight of all the dieters were taken and it was found that Group A had a standard deviation of 5 kg while Group B had a standard deviation of 9 kg. Can we conclude at the 1% level of significance that the variability of Group B is greater than that of A? Solution 74 Chapter 3 Test of Hypothesis Example 3.18 Factory A was suspected of illegally dumping the industrial waste into a river. To determine whether the factory was the culprit, random samples of water from 6 locations before the factory and 8 locations after the factory along the river where the dumping was suspected to be carried out. The amount of dissolved oxygen in water will decline if industrial waste was dumped into river. The results, in parts per million (ppm) were as follows: Before 5.3 5.5 5.1 5.4 5.7 5.0 After 4.2 5.3 4.4 4.0 5.4 3.8 3.9 4.3 Assuming a normal distribution for the amount of dissolved oxygen, a) Test at the 5% level of significance that the variability of the amount of dissolved oxygen is the same before and after the location of the factory. b) Using the results in part (a) as the assumption, test at the 5% level of significance that the data provides sufficient evidence to charge Factory A for illegal dumping of industrial waste into river. Solution 75 Chapter 3 Test of Hypothesis Example 3.19 Using the following output, is the difference in means statistically significant at 5 % level using a) a two-tailed test? b) a one-tailed test? t-Test: Two-Sample Assuming Equal Variances INDOOR OUTDOOR Mean 0.186429 0.623571429 Variance 0.004579 0.15099418 Observations 28 Pooled Variance 0.077787 28 Hypothesized Mean Difference 0 df 54 t Stat -5.86455 P(T<=t) one-tail 1.4E-07 t Critical one-tail 1.673566 P(T<=t) two-tail 2.81E-07 t Critical two-tail 2.004881 Solution 76 Chapter 3 Test of Hypothesis CHAPTER 3: EXERCISES 1. A sample consists of 36 glasses of orange juice was dispensed from a drink dispenser with a mean volume per glass of 21.9 ml and a standard deviation of 1.42 ml. The standard mean volume per glass of orange juice dispensed by the drink dispenser is 22.2 ml. Test the hypothesis that the mean volume per glass is less than the standard mean volume at =0.05. 2. A battery producer claimed that the mean lifetime of a battery produced by his factory was 30 hours. A competitor thought that the figure was too high and took 8 batteries at random and the lifetime of each is as follows: Battery A B C D E F G H Lifetime (hours) 15 28 33 24 28 25 31 27 Can the claim by the battery producer be accepted at the 5% significant level? 3. It was claimed that on an average a car could be driven at most 20,000 km per year. 100 cars were taken at random and the number of km per year driven was recorded and it was found that mean has increased to 23,500 km and the standard deviation was 3900 km. Test the hypothesis that mean number of km driven has increased at the 1% level of significance. 4. A sample of 8 cigarettes from a popular brand was tested for the tar contents and the mean volume of tar contained in a cigarette was 18.6 mg and the standard deviation was 2.4 mg. The company claimed that the mean volume of tar contained in its cigarettes was only 17.5 mg. Can the claimed be accepted at the 1% level of significance. 5. It was reported in a country that the mean age of all juveniles held in public custody in 1983 was 15.4 years. The mean age of 100 randomly selected juveniles currently being held in public custody is 15.26 years, and the 77 Chapter 3 Test of Hypothesis standard deviation of the ages is 1.01 years. Does it appear that the mean age of all juveniles has decreased since 1983? Perform the appropriate hypothesis test using = 0.01. 6. Hakim Pharmacy Limited, a pharmacy company claimed that not more than 1% that used its drug experience side effects. To prove its claim 2,500 patients were subscripted with the drugs and only 74 experienced the side effects. Can the claim by the Hakim Pharmacy Limited be accepted at the 5% level of significance? 7. A direct-mail firm wants to conduct a test-market for its new shampoo. A random sample of 800 people is chosen to receive advertising material describing the shampoo. It is decided that additional advertising and promotion will occur only if the sample results provide strong evidence that the actual response rate, p will exceed 6.5%. What decision will be made if 70 of the 800 people make a purchase? Use =0.01. 8. A new gadget is installed to air conditioner units in a hospital to minimize the number of bacteria floating in the air. The number of bacteria floating in the air before and after the installation is recorded as follows. Before installation After installation (No. of bacteria/ m 3 air) (No. of bacteria/ m 3 air) 10.1 12.8 11.6 8.2 12.1 11.6 9.1 14.1 10.3 15.9 15.3 9.0 13.0 14.5 78 Chapter 3 Test of Hypothesis Is it wise for the hospital management to install the new gadget? Test the hypothesis at the 1% level of significance. Assume the two population variances are equal. 9. The diameter of a screw is a critical measurement where it must be perfectly equal to a certain measurement. A machine is used to process screws with a diameter of 1 cm. If the diameters of screws are not equal to 1 cm, then the machine has to stop its production. A sample containing 36 screws is taken at random and the mean diameter is 1.006 cm and the standard deviation is 0.009 cm. Perform the appropriate hypothesis test at = 0.05 to determine whether the machine should stop its production. 10. A sample of mathematics marks from the two schools in Skudai is given as follows. Sekolah A: 78, 84, 81, 85, 81, 75, 79, 83, 76, 78 Sekolah B: 82, 87, 94, 88, 79, 80, 87, 83, 75, 85 With = 0.05, test the hypothesis that the mean performance of students in Sekolah A is better than that in Sekolah B. Assume the two population variances are the same. 11. Human IQ is normally distributed with a mean of 110 and a variance of 144. An educationist claims that university students have higher IQ than ordinary students. To prove his claim a sample of 30 university students are chosen at random and their mean IQ score is 114. Perform the appropriate hypothesis test at 5% significance level to prove the educationist claim. 12. A study is done to compare the height of students from the two selected schools in JB. 6 students from each school are chosen at random and the mean height and the standard deviation are recorded as follows: 79 Chapter 3 Test of Hypothesis Mean Variance Sekolah A 179.67 5.47 Sekolah B 181.00 5.40 Test the hypothesis that the mean height of students in Sekolah A is less than that in Sekolah B at the 5% level of significance. Assume the two population variances are equal. 13. A supplier of a new pesticide claims that its pesticide can increase the farmers’ production. A farmer volunteered to use the pesticide. 21 trees are chosen to use the pesticide and the mean and standard deviation of the production is 98 kg and 10 kg, respectively. Meanwhile, 16 trees are not treated with the pesticide and the mean and standard deviation of the production is 94 kg and 8 kg, respectively. Assume that the farmers’ production is taken from the same normal population. a) Find the 90% confidence interval for the mean production taken from trees that are treated with the pesticide. b) Find the 95% confidence interval for the ratio of two variances of the production taken from trees that used pesticides and trees that did not used pesticides. c) Test at 5% significance level whether the mean production of trees treated with the pesticide is more than the mean production of trees not treated by the pesticide. 14. The mean income per month for 30 workers chosen at random from the private sector is RM 650 with a standard deviation of RM 100 per month. The mean income per month for 40 workers from the government sector is RM 500 with a standard deviation of RM 80 per month. 80 Chapter 3 Test of Hypothesis a) Test the hypothesis that the mean income for all workers from government sector is less than RM 520 per month at 5% significance level. b) Test the hypothesis that the difference between the mean income of all workers from the private sector and the government sector is at least RM 100 at 5% significance level with the assumptions that (i) The two population variances are not equal. (ii) The two population variances are equal. c) Test the hypothesis that the ratio of the two population variances is equal to 1 at 5% significance level. 15. A sample poll of 300 voters from district X and 200 voters from district Y showed that 56% and 48%, respectively, were in favor of a given candidate. a) At a level of significance of 0.05, test the hypothesis that there is no difference between the districts b) Do the data provide sufficient evidence to support that the candidate is preferred in district X at the level of significance of 0.05. 16. A manufacturer claimed that at least 95% of the machine equipment he supplied to a factory conformed to specifications. An examination of a sample of 200 pieces of equipment revealed that 18 were faulty. Test his claim at a significance level of 0.01. 17. Random samples of 200 screws manufactured by machine A and 100 screws manufactured by machine B showed 19 and 5 defective screws respectively. Test the hypothesis that a) machine B is performing better than machine A b) the two machines are showing different qualities of performance at 5 % significance level. 81 Chapter 3 Test of Hypothesis 18. The standard deviation of the breaking strengths of certain cables produced by a company is given as 240 kg. After a change was introduced in the process of manufacturing of these cables, the breaking strengths of a sample of 8 cables showed a standard deviation of 300 kg. Investigate the significance of the apparent increase in variability using a significance level of 1%. 19. Two types of chemical solutions, X and Y were tested for their pH (degree of acidity of the solution). Analysis of 6 samples of X showed a mean pH of 7.52 with a standard deviation of 0.024. Analysis of 5 samples of B showed a mean pH of 7.49 with a standard deviation of 0.032. Assume the two population variances are equal. a) Using a 0.05 significance level, determine whether the two types of solutions have different pH values. b) Can we conclude that there is a significant difference in the variability of the pH values for the two solutions at a 0.01 level of significance? 20. An electronic company produces two types of components, namely X dan Y. A quality control officer would like to compare the lifetime of each component by taking samples from each. 13 samples are taken from X produce a mean of 14.2 hours and a standard deviation of 0.35 hours. 10 samples from Y are taken and the data is as follows: 13.6 a) 14.3 13.7 13.8 14.0 13.8 13.6 13.7 13.9 13.9 At = 0.1, show that the lifetime variances of both components X and Y are equal? b) If lifetime variances are equal, test at = 0.1 that X has longer lifetime mean than Y. 82 Chapter 3 Test of Hypothesis Chapter 3: Answers 1. z = -1.268 ; accept H0 2. t = -1.881 ; accept the claim 3. z = 8.974 ; 20,000 km 4. t = 1.296 ; accept the claim 5. z = -1.386 ; accept H0 6. z = 9.8494; reject the claim 7. z =2.5815; reject H0 8. t = -0.4899 ; accept H0 ; not wise to purchase the new gadget. 9. z = 4; reject H0 10. t = -1.989 ; reject H0 11.z = 1.8257; reject H0 12. sp = 2.331, t = -0.988; accept H0 13. a) (94.24 , 101.76) b) (0.566,4.016) 14. a) z = -1.5811; accept H0 c) accept H0 b)i. z = 2.2511; accept H0 ii.z = 2.324; accept H0 c) f = 1.5625; accept H0 15. a) z = 1.755; accept H0 b) z =1.755; reject H0 16. z = -2.5955; reject the claim 17. a) z = 1.496; accept H0 18. b) z =1.496; accept H0 test2 = 10.9375; not significant; accept H0 19. a) t = 1.7821; accept H0 b) f = 0.1927; not significant, accept H0 20. a) reject H0 b) reject H0. 83