# Calculating z-Scores, Normal Probabilities and Binomial Probabilities

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Problem 1:

In Module 2, you learned how to compute a z-score from a raw score. In this module, you are shown how to estimate the probability of getting a certain z-score value equal to or higher than the one that is observed (i.e., more extreme in the tail), as well as the proportion of all z-values that would NOT be in the tail of the distribution of all possible z-scores in a normally shaped distribution. To do this, you compute the z-score value and then look up the probabilities/proportions that match that z-score in Table B.1 in the back of your textbook.

Raw X Value Mean SD z-score Prop./Prob. in tail Prop./Prob. in body

A. 38.25 30 5

B. 39.80 30 5

C. 17.00 14 1.5

D. 11.5 14 1.5

Summary Which of the above raw score/z-values (a, b, c, and/or d) would be extreme enough to occur 5% or less of the time (i.e., p < .05) within its distribution of scores?

Problem 2:

An actual outcome can be compared with the probability of getting that outcome by chance alone. This is the basis of inferential statistics. In inferential statistics, we are comparing what we really observe with what would be expected by chance alone. That which would be expected by chance alone would be the null hypothesis (that is, nothing is going on here but chance alone).

If we were to throw a coin, there would be a 50% chance it would come up heads, and a 50% chance it would come up tails by chance alone. By extension, if we threw the coin 20 times, we'd expect 50% (p = .5 or pn = 20*.5 = 10) of the tosses to come up heads, and 50% (q = .5 or qn = 20*.5 = 10) to come up tails by chance alone if this is a fair coin.

a. You aren't sure if your friend is using a fair coin when he offers to toss the coin to decide who will win $100. You ask him to let you toss the coin 25 times to test it out before you decide whether you will take the bet, using this coin. You toss the coin 25 times and it comes up heads 19 times. Is this a fair coin (the null hypothesis)? What is the probability of getting 19 heads in 25 tosses by chance alone? You have decided that if the outcome of 19/25 tosses as heads would occur less than 5% of the time by chance alone, you will reject the idea that this is a fair coin.

b. Now, suppose the outcome of your trial tosses was 15 heads in 25 tosses. What is the probability of 15 heads in 25 tosses? Would you decide this is a fair coin, using the 5% criterion as in question a

Problem 3:

A teacher, Mrs. Jones, tests her 8th grade class on a standardized math test. Her class of 20 students (n) gets a mean score (M) of 80 on the test. She wants to know how her class did in comparison with the population of all 8th grade classes that have taken this test. She goes to a national database and finds out that the national average () of scores for the population of all 8th graders who took this test is 78, with a population standard deviation ( of 3 points.

a. Based on the population mean and standard deviation, what is the expected mean and standard deviation (standard error) for the distribution of sample means based on the sample size of 20 students in a class?

b. If this distribution of the sample means is normal, what would be the z-score equal to a mean test score of 80 that Mrs. Jones' class received?

c. When you look up the z-score you computed in part b, what is the probability of obtaining a sample mean greater than M = 80 for a sample of 20 in this population?

d. Mrs. Jones wants to know if her class did significantly better than the average 8th grade class on this test.

• What is the null hypothesis?

• What is the alternative hypothesis?

• Is the mean score obtained in Mrs. Jones' class (sample) significantly different from the population mean, using the criterion that her class's score would have to fall in the part of the distribution of all scores in the population that is above the mean and has frequencies of occurrence of 5% or less of all scores in the population (i.e., her class's mean score would have a probability of occurring by chance alone of p < .05)?

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This solution looks at how normal probabilities and binomial probabilities may be used to make a decision about a population parameter, preparing the way to understand hypothesis testing.

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Raw X Value Mean SD z-score1 Prop./Prob. ...

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