Tests of Hypothesis

A statistical hypothesis is an assertion or conjecture concerning one or more populations. There are two types of statistical hypotheses: the null hypothesis and the alternative hypothesis. The null hypothesis, denoted by Ho, is defined as a hypothesis of no difference. It is ordinarily formulated with the purpose of being accepted or rejected. If the null hypothesis is rejected, the alternative hypothesis is accepted. The alternative hypothesis denoted by H₁.

Alternative hypotheses are classified as either nondirectional or directional hypotheses. A nondirectional hypothesis is one which asserts that one value is different from another. More specifically, it is an assertion that there is a significant difference between two statistical measures (or that there are significant differences among three or more summary measures). A directional hypothesis, on the other hand, is an assertion that one measure is less than (or greater than) another measure of similar nature. Mathematically a nondirectional hypothesis make use of the “not equal to” (¹)sign, while a directional hypothesis involves one of the other relations, “less than” (<) or “greater than” (>). Nondirectional hypotheses are also called two-sided hypotheses, and directional hypotheses also known as one sided-hypotheses.

Consequences of Decisions in Testing Hypotheses

DecisionèFact	Ho is true	Ho is false
Accept Ho	Correct decision	TypeII error
Reject Ho	Type I error	Correct decision

One-tailed and two-tailed tests

A test of any statistical hypothesis where H₁ is directional called a one-tailed test. A test where H₁ is nondirectional is called a two-tailed test. The difference between these two types of tests lies in that the first is characterized by a region of rejection which lies entirely in one end of the distribution. Whereas the second involves a critical region which is split into two equal areas placed in each tail of the distribution.

Steps in Hypothesis Testing

<!--[if !supportLists]-->1. <!--[endif]-->Formulate the null hypothesis (Ho) and the alternative hypothesis(H₁).

<!--[if !supportLists]-->2. <!--[endif]-->Specify the level of significance a.

<!--[if !supportLists]-->3. <!--[endif]-->Choose the appropriate test statistic.

<!--[if !supportLists]-->4. <!--[endif]-->Establish the critical region.

<!--[if !supportLists]-->5. <!--[endif]-->Compute for the value of the statistical test.

<!--[if !supportLists]-->6. <!--[endif]-->Make a decision and if possible draw a conclusion.

Critical Value of Z

Significance Level	.10	.05	.25	.01
One-tailed	1.28	1.645	1.96	2.23
Two-tailed	1.645	1.96	2.33	2.58

<!--[if !supportLists]-->1. <!--[endif]-->An electrical company claims that the lives of the light bulbs it manufactures are normally distributed with a mean of 1000 hours and a standard deviation of 150 hours. What can you say about this claim if a random sample of 100 bulbs produced by the company has a mean life of 980 hours? Use a .05 level of significance.

<!--[if !supportLists]-->2. <!--[endif]-->An instructor gives his class an achievement test which, as he knows from years of experience, yields a mean 80. His present class of 40 obtains a mean of n85 and a standard deviation of 8. Can he claim that his present class is a superior class? Employ alpha .01.

<!--[if !supportLists]-->3. <!--[endif]-->A new production process is being considered to replace the old process presently used. This ne process was tested for 8 consecutive hours with the following results: 118, 122, 120, 124, 126, 125, 125, 124. If the average output per hour using the old process is 120 units. Is the management justified in stating that the output per hour can be increased with the new process? Use alpha .01

<!--[if !supportLists]-->4. <!--[endif]-->In a time and motion study, it was found that a certain manual work can be finished at an average time of 40 minutes with a standard deviation of 8 minutes. A group of 16 students is given a special training and then found to average only 35 minutes. Can we conclude that the special training can speed up the work using 0.01 level?

<!--[if !supportLists]-->5. <!--[endif]-->A standardized test was administered to thousands of students with a mean score of 85 and a standard deviation of 8. A random sample of 50 students were given the same test and showed an average score of 83.20. Is there evidence to show that this group has a lower performance than the ones in general at 0.05 level?

<!--[if !supportLists]-->6. <!--[endif]-->The average length of time for students to register for summer classes at a certain college has been 50 minues. A new registration procedure using modern computing machines is being tried. If a random sample of 35 students had an average registration time of 42 minutes with a standard deviation of 11.9 minutes under the new system, test the hypothesis that the population mean is now less than 50 minutes using a level of significance 0.01.

<!--[if !supportLists]-->7. <!--[endif]-->A university librarian suspects that the average number of books checked out to each student per visit has changed recently. In the past an average of 3.4 books was checked out. However, a recent sample of 45 students averaged 4.3 books per visit with a standard deviation of 1.5 books. At the 0.05 level of significance. Has the average checked out changed?

<!--[if !supportLists]-->8. <!--[endif]-->The Computer Center of a well known university has installed new color video display terminals to replace the monochrome units it previously used. The 32 graduating BS Computer Science students trained to use the new machines averaged 7.2 hours before achieving a satisfactory level of performance. Their sample standard deviation was 16.2 hours. Long experiences university computer operations on the old monochrome terminals showed that they averaged 8.1 hours on the machines before their performance were satisfactory. At the 0.10 significance level, should the head of the computer center conclude that the new terminals are easier to learn to operate?

<!--[if !supportLists]-->9. <!--[endif]-->In the certain studies of deception among the students the scores achieve on test given under conditions in which cheating was possible were compared with scores achieve by the sample students under strictly supervised condition. In a certain test given under “honest”conditions the mean score of 25 sample students is 62 and the standard deviation is 10. A student who took the test under non-supervised conditions turned in mean score of 87. What conclusion can be drawn out of the given information at 0.01 level of significance?

<!--[if !supportLists]-->10. <!--[endif]-->An educational researcher found that the average entrance score of the incoming freshmen in a famous university is 84. A random sample of 24 students from a public school was then selected has found out that their average score in the entrance examination is 88 with a standard deviation of 16. Is there any evidence to show that the sample from public school performed better than the rest in the entrance examination using the 0.10 level of significance?

<!--[if !supportLists]-->11. <!--[endif]-->A professor in a typing class found that the average performance of an expert typist is 85 words per minute. A random sample of 16 students took the typing test and an average speed of 62 words per minute was obtained with a standard deviation of 8. Can we say that the sample students performance is below the standard at the 0.05 level of significance?

<!--[if !supportLists]-->12. <!--[endif]-->A pharmaceutical firm claims that the average time for a drug to take effect is 18 minutes with a standard 0f 2 minutes. In a sample of 36 trials, the average time was 20 minutes. Test the claim against the alternative that the average time is not equal to 18 minutes, using a 0.01 level of significance.

<!--[if !supportLists]-->13. <!--[endif]-->In a time and motion study, it was found out that the average time required by workers to complete a certain manual operation was 26.6 minutes with a standard deviation of 3 minutes. A group of 25 workers was randomly chosen to receive a special training for 2 weeks. After the training, it was found that their average time was 24 minutes. Can it be concluded that the special training speeds up the operation? Use alpha 0.05

<!--[if !supportLists]-->14. <!--[endif]-->A ceratain brand of powdered milk is advertised as having a net weight of 250 grams. If the net weights of a random sample of 10 cans are 253, 248, 252, 245, 247, 249, 251, 250, 247 and 248 grams, can it be concluded that the average net weight of the cans is less than the advertised amount? Use alpha 0.01

<!--[if !supportLists]-->15. <!--[endif]-->The mean content of 25 bottles of brand S mango juice is 335 mL with a standard deviation of 9 mL. Is this in line with the manufacturer’s claim that the bottle contains on the average 360 mL? Use a 0.01 level of significance.

<!--[if !supportLists]-->16. <!--[endif]-->The daily wages a particular industry are normally distributed with a mean of P66. If a company in this industry employing 144 workers pays, on the average, P62 with a standard deviation of P 12.50, can this company be accused of paying inferior wages at the 0.01 level of significance?

<!--[if !supportLists]-->17. <!--[endif]-->The manager of an appliance store, after noting that the average daily sales was only 12 units, decided to adopt a new marketing strategy. Daily sales under this strategy were recorded for 90 days after which period the average was found to be 15 units with a standard deviation of 4 units. Does this indicate that the new marketing strategy increased the daily sales? Employ alpha 0.01. Assume that the distribution of daily sales are approximately normally distributed.