Normal Distribution and Inferential Statistics
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Normal Distribution and Inferential Statistics
Lisa F. Peters
University of Phoenix
Normal Distribution and Inferential Statistics
5.10 Exercises 2,8,14 and 20
2. Using the normal curve table, determine the area of the standard normal distribution that is less than the following z-scores:
a. z = .30
b. z = 1.75
c. z = 2.42
d. z = −.68
e. z = −1.11
8. What percentage of IQ scores is ...
a. between 115 and 130
b. between 105 and 120
c. between 75 and 85
d. between 80 and 95
e. between 90 and 110
f. between 80 and 120
14. How many base hits can a baseball team expect to get in a game? Frohlich (1994) recorded the number of hits by the 28 major league baseball teams for all of the games played from 1989 to 1993 (each team plays 162 games a year). He found that the number of hits the teams made in the games was normally distributed, with a mean of 8.72 and a standard deviation of 1.10.
a. In what percentage of games would you expect a team to get more than nine hits?
b. In what percentage of the games would you expect a team to get less than six hits?
20. You collect the grade point average (GPA) from 10 students:
a. Calculate the mean (x) and standard deviation (s) of these GPAs.
b. Calculate the z-score for each GPA in order to standardize the distribution.
c. Calculate the mean and standard deviation of these z-scores.
6.9 Exercises 2,6,16 and 20
2. You have collected the following data:
6 4 3 7 4 2 6 5 7 4
If you randomly select one of these 10 numbers, what is the probability the number (X) will be...
a. equal to 4?
b. equal to 7?
c. less than 5?
d. greater than 2?
e. greater than 4 but less than 7?
6. According to the test's publishers (www.act.org), scores on the ACT college entrance examination for students graduating in 2001 were normally distributed, with μ = 21 and σ = 5 (scores can range from 1−36).
a. What is the probability of having a score greater than 30?
b. What is the probability of having a score greater than 20?
c. What is the probability of having a score less than 17?
d. What is the probability of having a score less than 23?
e. What is the probability of having a score between 18 and 25?
16. For each of the following situations, state the two competing hypotheses to be tested using words rather than mathematical symbols or formulas.
a. A company designs a program aimed at helping people stop smoking. They design a study aimed at testing the program.
b. A study wished to examine whether using seat belts affected the severity of injuries sustained by children in automobile accidents (Osberg & Di Scala, 1992).
c. A researcher hypothesizes that the more drivers use cellular phones, the greater the likelihood of getting into a traffic accident.
d. "It was hypothesized that ... infants who spent greater amounts of time in center-based care would demonstrate more advanced exploratory behaviors than infants who did not spend as much time in center-based care" (Schuetze et al., 1999, p. 269).
e. "It is expected that achievement motivation will be a positive predictor of academic success" (Busato et al., 2000, p. 1060).
f. "The purpose of our study was to gain a better understanding of the relationship between social functioning and problem drinking....We predicted that problem drinkers would endorse more social deficits than nonproblem drinkers" (Lewis & O'Neill, 2000, pp. 295−296).
20. Two researchers test the same research hypothesis using the same instruments. One researcher rejects the null hypothesis but the other does not.
a. Which researcher is more likely to have had a larger sample size? Why?
b. Which researcher is more likely to have had a smaller level of alpha? Why?
c. Which researcher is more likely to have had a directional alternative hypothesis? Why?
References
Tokunaga, H. T. (2016). Measures of central tendency. Fundamental Statistics for the Social and
Behavioral Sciences. SAGE Publications, Chptr 5 and 6.
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Solution Summary
The solutions are provided in the Word file. Additional computations are provided in the Excel file.
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5.10
2) Look up the numbers in a normal curve table such as the one shown on the previous page.
a) P (z < .30) = 0.5 + 0.1179 = 0.6179
b) P (z < 1.75) = 0.5 + 0.4599 = 0.9599
c) P (z < 2.42) = 0.5 + 0.4922 = 0.9922
d) P (z < −.68) = 0.5 - 0.2517 = 0.2483
e) P (z < −1.11) = 0.5 - 0.3665 = 0.1335
8) IQ scores have a mean of 100 and sd of 15. Use the table to look up the numbers in the computations for z.
a)
z1 = (115 - 100) / 15 = 1
z2 = (130 - 100) / 15 = 2
P (1 < z < 2) = 0.4772 - 0.3413
= 0.1359
=13.59%
b)
z1 = (105 - 100) / 15 = 0.33
z2 = (120 - 100) / 15 = 1.33
P (0.33 < z < 1.33) = 0.4082 - 0.1293
= 0.2789
= 27.89%
c)
z1 = (75 - 100) / 15 = -1.67
z2 = (85 - 100) / 15 = -1
P (-1.67 < z < -1) = 0.4525 - 0.3413
= 0.1112
= 11.12%
d)
z1 = (80 - 100) / 15 = -1.33
z2 = (95 - 100) / 15 = -0.33
P (-1.33< z < -0.33) = 0.4082 - 0.1293
= 0.2789
= 27.89%
e)
z1 = (90 - 100) / 15 = -0.67
z2 = (110 - 100) / 15 = 0.67
P (-0.67< z < 0.67) = 0.2486 + 0.2486
= 0.4972
= 49.72%
f)
z1 = (80 - 100) / 15 = -1.33
z2 = (120 - 100) / 15 = 1.33
P (-1.33< z < 1.33) = 0.4082 + 0.4082
= 0.8164
= 81.64%
14) Given mean = 8.72, sd = 1.10. There are a few ways this problem could be done. Different methods give different answers. Since the file is entitled Normal Distribution and Inferential Statistics (with no mention of the Binomial Distribution, correction for continuity, or anything else), I will do the problem based on the method the title most closely suggests.
a)
z = (9 - ...
Education
- MSc, California State Polytechnic University, Pomona
- MBA, University of California, Riverside
- BSc, California State Polytechnic University, Pomona
- BSc, California State Polytechnic University, Pomona
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