# A Step-by-Step Introduction to Statistics for Business

# Chapter 4: Probability Distributions

Answers for **Data Skill Challenges** for every chapter in the book can be found to check your performance and widen your understanding.

1) To answer these questions, you must first compute the mean and standard deviation of this dataset. The mean is 2.857143. The standard deviation is 1.345185. Remember to carry these out to six decimal places, since we’ll need them in later calculations.

1a) −1.38, 0.11, −0.64, 1.59, 0.85, 0.11, −0.64

1b) The *z*-score associated with 2.5 is −0.27. The proportion of values in the tail associated with this *z*-score is .3936. Since the *z*-score is negative, this value is in the lower tail. So if we want to know what proportion of cases is *above *this point, we must calculate 1 −.3936 = .6064. The question asks for a percentage – not a proportion – so we must then convert this to 60.64%.

1c) The 20th percentile corresponds to *z* = −.84. Converting this to a raw score gives us 1.73.

2) To answer these questions, you must first compute the mean and standard deviation of this dataset. The mean is 3. The standard deviation is 1.632993. Remember to carry these out to six decimal places, since we’ll need them in later calculations.

2a) 0, 1.84, –0.61, –1.22, –.61, 0, .61

2b) The *z*-score associated with 2 is –0.61. The proportion of values in the tail associated with this *z*-score is .2709. Since the *z*-score is negative, this value is in the lower tail. Since this number represents the proportion of cases below this point, we can use this value to answer the question. The question asks for a percentage – not a proportion – so we must then convert this to 27.09%.

2c) The 90th percentile corresponds to *z* = 1.28. Converting this to a raw score gives us 5.09.

3) To answer these questions, you must first compute the mean and standard deviation of this dataset. The mean is 4.285714. The standard deviation is 1.603567. Remember to carry these out to six decimal places, since we’ll need them in later calculations.

3a) 45,.45, 1.69, −1.43, −.80, −.18, −.18

3b) The *z*-score associated with 4 is −0.18. The proportion of values in the tail associated with this *z*-score is .4286. Since the *z*-score is negative, this value is in the lower tail. So if we want to know what proportion of cases is *above *this point, we must calculate 1 −.4286 = .5714. The question asks for a percentage – not a proportion – so we must then convert this to 57.14%.

3c) The 75th percentile corresponds to *z* = .67. Converting this to a raw score gives us 5.36.

4) To answer these questions, you must first compute the mean and standard deviation of this dataset. The mean is 2.571429. The standard deviation is .9759. Remember to carry these out to six decimal places, since we’ll need them in later calculations.

4a) –.59, .44, .44, –1.6, 1.45, –.59, .44

4b) The *z*-score associated with 3 is .44. The proportion of values in the tail associated with this *z*-score is .3300. Since the *z*-score is negative, this value is in the lower tail. Since this number represents the proportion of cases below this point, we can use this value to answer the question. The question asks for a percentage – not a proportion – so we must then convert this to 33%.

4c) The 40th percentile corresponds to *z* = –.25. Converting this to a raw score gives us 2.33.