Chapter 12. The One- Sample

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1 Chapter 12 The One- Sample z-test

2 Objective We are going to learn to make decisions about a population parameter based on sample information.

3 Lesson Testing a Two- Tailed Hypothesis

4 Example 1: Let's imagine twenty years from now you are the dean of the graduate school of psychology at De La Salle College of Saint Benilde-Antipolo Campus. Unlike many other graduate schools in the United States, your students are not required to take the Graduate Record Examination (GRE) as part of the admission process. Critics have suggested that this attracts weaker students and, because of that they will not score as high on the national board certifications as their peers in schools where the GRE is required.

5 Example 1: You admit to yourself that you've never really given it too much thought and, to tell the truth, you're really not sure how your students do in comparison to students in other graduate schools. Interested in finding out, you visit one of the statistics professors in your department and tell her what's going on.

6 Example 1: After telling her your problem, your professor immediately tells you "In order to conduct your study, you have to remember two things."

7 First: "our student body is assumed to be nothing more than a sample of the population of all graduate psychology students throughout the Philippines. There are an infinite number of other samples you could select from that sake population."

8 Second: "What you want to know is if our students are really members of that population or, because of our admissions process, they are significantly different from other samples from that population. In other words, you want to know if the average score our students make on national certification examination is significantly different from the average score of the other samples."

9 Example 1: "As you can see, our critics are claiming that our students will not do as well on the certification exams as the other schools, and we do not know if we are doing significantly better or worse than they are. Given that, you need to develop the hypothesis you want to test and then collect the data you need."

10 Two-Tailed Research Hypothesis: "There will be a significant difference in scores on the national certification exam between graduate psychology students at De La Salle College of Saint Benilde-Antipolo Campus and graduate psychology students at other universities."

11 Questions: What kind of hypothesis is this? Why do you think it is a two-tailed research hypothesis? Where does two-tailed pertaining to? What will be the null hypothesis?

12 Null Hypothesis: "There will be no significant difference in scores on the national certification exam between graduate psychology students at De La Salle College of Saint Benilde-Antipolo Campus and graduate psychology students at other universities."

13 Question: Why do we need to make our hypothesis twotailed?

14 Chapter 12 Four Good Things When Testing a Hypothesis of a Normally Distributed Sample

15 Four Good Things When a Hypothesis of a Normally Distributed Sample 1. We can test hypothesis to determine if one of the sample means is significantly different from the population mean by using the z score we have already learned to compute. We call this a one-sample z test.

16 Four Good Things When a Hypothesis of a Normally Distributed Sample 2. We saw earlier that the z score allowed us to report distances from the mean in terms of the standard deviation. The onesample z-test allows us to do the same but, instead of using the standard deviation, we will use the standard error of the mean.

17 Four Good Things When a Hypothesis of a Normally Distributed Sample 3. The empirical rule still applies.

18 Four Good Things When a Hypothesis of a Normally Distributed Sample 4. The principles underlying the z-test are fairly straightforward and are basically the same for most of the other statistical tests we will use.

19 Going Back: Let's suppose we have 100 students in our graduate program. Let's assume the population mean is 800 (i.e., the average score for graduate students at all universities in the Philippines), the SEM is 10, and the mean value for our students is 815.

20 Formula for Computed Value of z:

21 Where: 1. The mean score we want to compute for a z score for. 2. Either the population mean or the mean of means. 3. The population standard error of the mean ( ).

22 Computed Value of z It is the value of z or the z statistic we computed to test our hypothesis.

23 Critical Value of z It is nothing more than a value from the area under the normal curve table where we have to compare our computed value of z to know if significant difference exists.

24 Chapter 12 Steps in Determining the Critical Value of z

25 Step 1: We have to decide on our alpha values. Here, we can use the traditional.05 alpha value (i.e., 5%).

26 Step 2: We have to use the type of hypothesis we have stated to determine the distribution of all of the possible critical values of z.

27 Note 1: If we have a twotailed hypothesis, we have to distribute the z scores equally under the curve.

28 Note 2: If we have a onetailed hypothesis we have to distribute them according to the direction of the hypothesis.

29 Question: What are we going to do with a range that includes 95% of all possible z values? How about the left over which is our alpha value 5%?

30 Step 3: Determine exactly which value we want to use as our critical value of z using the area under the normal curve table.

31 Note: Since we are testing a nondirectional hypothesis, a two-tailed, hence this means, for 47.5%, our critical value of z on the right side of the mean is and the critical z value on the left side of the mean is. Thus, we will have a range of critical z score from.

32 Now, Testing the Two-Tailed Hypothesis: How to do it? All we have to do to determine if our group is significantly different is to see if our computed z score falls within 95% range of all possible critical z scores.

33 Final Answer: Our computed value of z is 1.5 and our range of critical value of z is from, and 1.5 is between them. This means we fail to reject our null hypothesis; there is no significant difference between our scores and the scores of the other universities throughout the nation.

34 Example 2: Let's use the national mean score of 800 again, but this time let's use a mean score of 780 and a SEM of 10. (Solution on the Board)

35 Final Answer: This gives us a z score of - 2. Since this doesn't fall between, we will reject the null hypothesis. In this case, our score of 780 is significantly less than the national average of 800

36 Question: What is the basis of saying it is less than and not greater than? As the dean, what action will you now implement?

37 Lesson Testing a "Greater Than" One-Tailed Hypothesis

38 Question: Where does one-tailed refer to?

39 Answer: Here, we will be using our alpha value on only one side of the distribution. Since we are testing onetailed hypothesis we will be dealing with a directional hypothesis.

40 Example 3: Imagine you're the dean of a school of education where the average professor's monthly salary is P73, One day, one of your more outspoken faculty members comes into your office and asks, "How do you ever expect us to stay here? We work harder, teach more classes, and have a larger dissertation load than faculty at other universities. Unless you can show us we are making significantly more than the average faculty salary at those other places, we are going on strike!" After saying this, the professor leaves your office in a huff, slamming the door behind him. After careful consideration, you decide the professor might have a point. He is pretty good and he does a lot of fine work; perhaps an investigation is called for.

41 Question: So, what will you do now? What hypothesis can you make?

42 Research Hypothesis: "Your faculty's average monthly salary will be significantly higher than the national average salary of faculty members."

43 Null Hypothesis: "There will be no significant difference between your faculty's average monthly salary and the national average salary of faculty members."

44 Going Back: In order to begin investigating this, you call the human resources department and find that the average salary in your department is P74, After a little investigation, they tell you the national average is P70, with a SEM of P2,

45 Question: What will be the computed value of z? (Solution on the Board)

46 Final Answer: z = +2

47 Question: Is that large enough to consider the difference significant?

48 Determine the Critical Value of z Question: What did we do when we were using a two-tailed hypothesis?

49 Determine the Critical Value of z In this case we have to do something a bit different. Since we have a one-tailed hypothesis but still have an alpha value of.05, we have to mark the entire 5% off on one end of the distribution or the other.

50 Question: What is the direction of our hypothesis?

51 Answer: Here, we have a "greater than" hypothesis so we have to mark the 5% on the positive end of the distribution.

52 Task: Find the critical value using area under the normal curve table.

53 Answer: The critical value of z is

54 Question: Is our computed z greater than our critical value of z? What will be our general decision?

55 Answer: Since it is greater than the critical value, we will reject the null hypothesis. Your faculty members are making, on average, P74, per month. This is significantly more than the national average of P70,

56 Lesson Testing a Less Than" One-Tailed Hypothesis

57 Example 4: Let's take a look at a "less than" case by using an average monthly faculty salary of P66, Since we already know the national average is P70, with SEM of P2,000.00, we are concerned that the faculty may have a justifiable complaint.

58 Question: From this, what will be your research hypothesis?

59 Research Hypothesis: "Your faculty's average monthly salary will be significantly less than the national average salary of faculty members."

60 Research Hypothesis: "Your faculty's average monthly salary will be significantly less than the national average salary of faculty members."

61 Null Hypothesis: "There will be no significant difference between your faculty's average monthly salary and the national average salary of faculty members."

62 Task: Compute for the computed value of z.

63 Question: From the computed value of z, what our distribution should look like?

64 Answer: Since we are looking at a "less than" relationship, we have to put the 5% on the left (i.e., negative) side of the distribution.

65 Question: What is our critical value of z? What decision can you make after comparing our computed value of z to critical value of z?

66 Answer: If we compare our computed value of z (i.e., -1.75) to our critical value of z, it is less than Because of that, the professor seems to have a case; the faculty's average monthly salary is significantly lower than the national average.

67 Question: As the dean, what action will you make if the computed value of z is -3? How about if -1?

68 Danger: Be Careful When Changing Your Alpha Values While we do usually use an alpha value of.05, there are instances where we might want to change. When we do, we have to think back to the idea of Type I and Type II errors.

69 Danger: Be Careful When Changing Your Alpha Values

70 Question: From the table, what happens when we test a hypothesis and use a larger alpha value (i.e.,.10)?

71 Answer: When we test a hypothesis and use a larger alpha value, we create a narrower confidence interval. This means we're actually creating a smaller range of values that we would consider not significantly different from the value we're interested in.

72 Question: When our confidence interval is narrower, how does it affect our decision on whether to reject or fail to reject our hypothesis?

73 Answer: Anything outside of that range would be considered significantly different and would cause us to reject our null hypothesis. A larger alpha gives us a far better chance of rejecting the null hypothesis. Unfortunately, this also means that we're increasing our Type I Error rate; we may be rejecting a null hypothesis when we shouldn't.

74 Question: From the table, what will happen if we decrease our alpha value to.01?

75 Answer: We're greatly widening the range of values we would consider not significantly different from our mean score.

76 Question: When our confidence interval is wider, how does it affect our decision on whether to reject or fail to reject our hypothesis?

77 Answer: This, of course, lowers our probability of rejecting the null hypothesis and greatly increases the probability of a Type II error; we might fail to reject the null hypothesis when we actually should.

78 Final Remark: Having too large or too small of an alpha value creates problems; a good consumer of statistics will usually use an alpha value of.05 and its acceptable probability of making either a Type I or Type II error.

79 Question: What do we know about Inferential Statistics? What is the heart of Inferential Statistics?

80 One-Sample z Test It is testing a hypothesis about population mean using information from both the population and the sample (only one sample).

81 Lesson 12.2 The Probability Values or p-value

82 p-value It is the probability that a particular outcome is due to chance. In the case we just discussed about the salaries of universities, another way of describing the is to call it the probability of our university's salaries being equivalent to the salaries from other universities.

83 Note: These range from 0.00 (no probability that the same mean came from the population being considered) to 1.00 (an absolute certainty that the sample mean came from the population being considered).

84 Lesson 12.2 Steps in Computing for p- value

85 Step 1: Compute the value of z.

86 Step 2: Construct a normal distribution and plot the computed value of z.

87 Step 3: Determine the area under the normal curve of your computed value of z and subtract it to

88 Step 4: Multiply the difference by 2 to get the p value.

89 Step 5: Compare the p value to the alpha value.

90 Note: If we are trying to reject the null hypothesis, we want a small p value as possible. This will help us ensure that any differences we find are "real" and not due to chance.

91 Note: If we are trying to reject the null hypothesis, we want a small p value as possible. This will help us ensure that any differences we find are "real" and not due to chance.

92 Lesson 12.2 Comparing p value to the Predetermined Alpha Value

93 Conclusion 1: If the computed p value is less than the alpha value, a researcher will reject the null hypothesis; this means the differences are significant and not attributable to chance.

94 Conclusion 2: If the p value is greater than or exactly equal to the alpha value, the statistician will fail to reject their null hypothesis. This means any differences found are not significant or are attributable only to chance.

95 Direction: Look at the following hypotheses and their alpha and p values. Based on them, let's decide whether we need to reject or fail to reject our null hypothesis.

96 Example 5: Null Hypothesis: "There will be no significant difference in achievement between students who eat breakfast and those who do not eat breakfast."

97 Example 5: Suppose we have collected achievement scores for a random sample of students and then asked them to tell us whether or not they eat breakfast. After using a computer to calculate the descriptive statistics, we have found the average score for students eating breakfast is 82 while that of students skipping breakfast is 79.

98 Question: Based on the information (mean scores), what immediate conclusion can you draw?

99 Example 5: Using our standard alpha value of.05, let's assume the output from that procedure shows a p value of.05.

100 Question: Now, what final decision can you make?

101 Answer: In this case, p value is exactly equal to alpha value, so we will fail to reject the null.

102 Question: From your hypothesis, what conclusion can you make?

103 Answer: Even though the mean scores are different, apparently eating breakfast does not significantly affect achievement in school one way or another.

104 Example 6: Null Hypothesis: "There will be no significant difference in scores on the Beck Depression Inventory between patients receiving behavioral therapy and those receiving cognitive therapy."

105 Example 6: Null Hypothesis: "There will be no significant difference in scores on the Beck Depression Inventory between patients receiving behavioral therapy and those receiving cognitive therapy."

106 Example 6: Let's suppose we have a pool of depressed patients. We randomly select one group to receive cognitive therapy and the other to receive behavioral therapy. After an appropriate period of time, say 10 sessions with the psychologist, we ask each of the patients to complete the Beck Depression Inventory. After computing the descriptive statistics, we see that the cognitively treated patients have a mean score of 15 while behaviorally treated patients have a mean score of 30.

107 Question: From the given mean scores, what can you say to cognitively treated patients and to behaviorally treated patients? Which among them is effective?

108 Example 6: Let's use our alpha value of.05, and, using the data we collected, our software computes a p value of.01.

109 Question: Now, what final decision can you make?

110 Answer: Since this is less than our alpha value of.05, obviously the cognitive therapists know something the behaviorists do not; their clients' scores are significantly less than those of the behaviorists' clients.

111 Example 7: Null Hypothesis: "There will be no significant difference in elementary school achievement between students attending public school and those being home-schooled."

112 Example 7: Let's use the results of a standardized test that the students take at the end of each school year. The results show a mean score of 56 for the public school students and 62 for the home-schooled students. This time, let's set alpha equal to.10 and find that p is equal to.11.

113 Question: Based on the given, what final decision can you make?

114 Answer: In this case, p is greater than our alpha value, so we fail to reject our null hypothesis. We support our research hypothesis.

115 Question: From your hypothesis and result, what can you conclude?

116 Answer: It is apparent that it makes no difference where students attend elementary school -- they all do equally as poorly on the standardized tests!

117 p value and alpha value

118 Lesson 12.3 Choosing the Right Statistical Test

119 Lesson 12.3 What We Already Knew in Choosing the Right Statistical Test

120 What We Already Knew If the data we collected for our dependent variable are either nominal or ordinal, we will use nonparametric procedures.

121 What We Already Knew If the data are quantitative (interval or ratio level), most of the time we will use a parametric test.

122 What We Already Knew In rare instances, if we have quantitative data that are not normally distributed, we will use a nonparametric procedure.

123 Lesson 12.3 Things to Consider in Choosing the Right Statistical Test

124 Things to Consider in Choosing the Right Statistical Test The type of data we are collecting.

125 Things to Consider in Choosing the Right Statistical Test The number of independent variables.

126 Things to Consider in Choosing the Right Statistical Test The number of levels of each of the independent variables.

127 Things to Consider in Choosing the Right Statistical Test The number of dependent variables.

128 Things to Consider in Choosing the Right Statistical Test

129 Question 1: Why is it not logical to consider either the number of independent variables or their levels for one-sample z-test or one-sample t-test?

130 Question 2: Why there are no alternative statistical tests to others on the table?

131 Question 3: Why we do not have independent and dependent variables in correlation procedures?

132 Let s Practice: Directions: Look at tables below. In each, you'll see a null and research hypothesis, data values for the population and sample, an alpha value, and the standard error of the mean. Use these values to compute the appropriate z score, obtain the critical z score, and then determine whether or not you should reject the null hypothesis. Finally, based on everything, determine if the p value would be less than.05. Again, you don't have to compute a p value; just state whether or not it would be less than.05 based on whether or not you would reject the null hypothesis based on the other computed values. In the last line of the table, support your decision about the p value.

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