The Wechsler IQ test is normally distributed and has a known μ = 100 and a σ = 15.
A sample of n = 25 is drawn at random.
A. The probability is .04 that the mean of the sample will be below what value?
__________________
B. What middle range of sample IQs will be expected to occur 95% of the time?
___________________
• In the first set of “Your Turn!” questions that follow, you are given a sample mean and are asked for a proportion or probability. This requires that the sample mean be changed into a z-score so that you can use the z-table to find the proportion or probability of interest:
z M
M
• In the second set of questions, you are given a probability or percentage and are asked for the value of a sample mean (or means). This requires first looking in the z-table to locate the z-score associated with the given probability or percentage and then using the formula for finding the value of the mean:
M z M
Answers to “Your Turn!” Problems I. Finding Probability or Proportion of Given Sample Means
A.
M n 12
36 2 z M
M
99 100
2 .50 p M 99.6915 B. z M
M
95 100
2 2 50. p M( 95) .0062 FORMULA SUMMARY
The formulas that you were introduced to in this chapter are summarized below:
z X u
tells how much a particular raw score deviates from the mean of a population in standard deviation units.
z M u
M
tells how much a particular sample mean deviates from the population in standard error units.
X = μ + (z)(σ) – use when being asked for a raw score value.
M = μ + (z)(σM) – use when being asked for the value of a sample mean.
Additional Practice Problems
Answers to odd numbered problems are in Appendix C at the end of the book.
1. What is a sampling distribution and what is its purpose?
2. Briefly discuss the characteristics outlined by the central limit theorem.
3. Why is it important to exercise caution when using a sample mean to make a prediction about the population mean?
4. The Wechsler IQ has a μ = 100 and a σ = 15.
a. Compute σM for a sample size of n = 120.
b. Compute σM for a sample size of n = 50.
c. How is σM affected by sample size?
5. Explain what is meant by the standard error of the mean. How is it related to sample size?
What are the practical implications of this relationship?
6. Given a normally shaped population distribution with μ = 88 and σ = 12, what is the probability that an obtained sample of size n = 36 will have a mean that is
a. above 92?
b. above 87?
c. below 87?
7. An assessment instrument on assertiveness has been standardized with a mean of 90 and a standard deviation of 20. A large manufacturing company administers the test to its sales force. If n = 64, determine
a. p(M < 92) b. p(M < 86) c. p(M < 96)
8. Given a normal population distribution with μ = 62 and σ = 15,
a. what is the probability of obtaining a sample mean less than 61 if n = 53?
b. what proportion of sample means can be expected to score above 59 if n = 36?
II. Finding Sample Means from Given Probabilities or Percentages
A.
M n 15
25 3 M z M
( )( ) ( . )( ) .
100 1 75 3 94 75
B. Need to identify the extreme 5%, 2ẵ% at each end. The z-score associated with .0250 is ±1.96. Thus,
M z M
( )( ) ( . )( ) .
100 1 96 3 94 12
M= + z M
= + +
=
à ( )(σ ) ( . )( ) .
100 1 96 3 105 88
Answers to “Your Turn!” Problems (continued)
9. Given a normal population distribution with μ = 71 and σ = 14,
a. what is the probability of obtaining a sample mean greater than 76 if n = 44?
b. what proportion of sample means can be expected to be below 76 if n = 24?
10. A population is normally distributed with μ = 36 and σ = 8. A sample of n = 25 is drawn at random.
a. The probability is .05 that the sample mean will be above what value?
b. The probability is .01 that the sample mean will be below what value?
11. A random sample of size n = 63 is drawn from a normally distributed population with μ = 106 and σ = 18.
a. The probability is .10 that the mean of the sample will be below what value?
b. The probability is .04 that the mean of the sample will be above what value?
12. A population distribution is normally shaped with μ = 83 and σ = 13. If samples of size n = 36 are drawn at random, what range of sample means would be expected to occur in the middle of the distribution 90% of the time?
13. A normally shaped population distribution has a μ = 73 and a σ = 11. If samples of size n = 22 are drawn at random, what range of sample means would be expected to occur in the middle of the distribution 95% of the time?
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HYPOTHESIS TESTING
Now that we have examined the properties of the normal distribution and you have an understanding of how probability works, let us look at how it is used in hypothesis test- ing. Hypothesis testing is the procedure used in inferential statistics to estimate population parameters based on sample data. The procedure involves the use of statistical tests to deter- mine the likelihood of certain population outcomes. In this chapter, we will use the z-test, which requires that the population standard deviation (σ) be known.
8
Hypothesis Testing
The material in this chapter provides the foundation for all other statistical tests that will be covered in this book. Thus, it would be a good idea to read through this chapter, work the problems, and then go over it again. This will give you a better grasp of the chapters to come.
Tip!
Hypothesis testing usually begins with a research question such as the following:
Sample Research Question
Suppose it is known that scores on a standardized test of reading comprehension for fourth graders is normally distributed with μ = 70 and σ = 10. A researcher wants to know if a new reading technique has an effect on comprehension. A random sample of n = 25 fourth grad- ers are taught the technique and then tested for reading comprehension. A sample mean of M = 75 is obtained. Does the sample mean (M) differ enough from the population mean (μ) to conclude that the reading technique made a difference in level of comprehension?
Our sample mean is, obviously, larger than the population mean. However, we know that some variation of sample statistics is to be expected just because of sampling error. What we want to know further is if our obtained sample mean is different enough from the popula- tion mean to conclude that this difference was due to the new reading technique and not just to random sampling error.
HYPOTHESIS TESTING STEPS
The process of hypothesis testing can be broken down into four basic steps, which we will be using throughout the remainder of this text:
1. Formulate hypotheses.
2. Indicate the alpha level and determine critical values.
3. Calculate relevant statistics.
4. Make a decision and report the results.
Let us examine each of these steps separately and apply them to our research question.