Test 4 Practice Test 1 For questions 1 through 4 write the appropriate answer on the line 1 Z has a standard normal distribution What is the probability that Z equals 1 57 2 What is the standard deviation of a standard normal distribution 3 Which of the following are correct comparisons between the Z distribution and all t distributions List the letters of all answers that are correct a They have the same mean b They have the same unusual features c They have the same standard deviation d They have the same shape 4 A variable X has a normal distribution with mean 54 and standard deviation 5 Approximately what percentage of the observations will fall between 44 and 64 Dr Pepper is a carbonated soft drink marketed as having a unique flavor Of interest is the quantity of Dr Pepper measured in ounces that a person consumes when they drink Dr Pepper served from a fountain machine Suppose that the quantity of Dr Pepper consumed follows an approximate normal distribution with mean 34 ounces and standard deviation 5 8 ounces Use this information to answer questions 5 and 6 5 Antravious purchased a Dr Pepper fountain drink and drank 41 ounces Based on the distribution described above what is the probability that the quantity of Dr Pepper consumed is 41 ounces or less 6 Shevani does not like soft drinks and hence consumes very little Dr Pepper Based on the distribution described above how many ounces of Dr Pepper would a person need to consume to fall in the bottom 4 01 of the distribution Suppose the ages of employees at the Dr Pepper Snapple Bottling Company follow an approximate normal distribution with a mean of 42 2 years and a standard deviation of 7 8 years Use this information to answer questions 7 8 and 9 7 Chnar and Rommy both work for the Dr Pepper Snapple Bottling Company Chnar is 30 years old while Rommy is 51 years old Based on the distribution described above what is the probability that the age of an employee will be between 30 and 51 years old 8 Racheed is one of the oldest employees at the Dr Pepper Snapple Bottling Company Based on the distribution described above how old must an employee be to fall in the top 7 35 of the distribution 9 Char Rese is 22 years old and works at the Dr Pepper Snapple Bottling Company Based on the distribution described above what is the probability that the age of an employee will be 22 years or higher Suppose we have a t distribution with 26 degrees of freedom Use this to answer questions 10 and 11 10 Find the probability that t26 is greater than 1 058 11 Find the value of t26 such that the probability of being less than that value is 995 Test 4 Practice Test 1 Solutions 0 zero The probability that a normal variable is exactly equal to any value is zero 1 one The standard deviation of the standard normal Z distribution is one A B and D The Z distribution and all t distributions all have a normal shape a mean of zero and no unusual features The only difference between the Z and all t distributions is that they have different standard deviations 95 Using the 68 95 99 7 rule we know that 95 of the data falls within two standard deviations of the mean Since the mean is 54 and the standard deviation is 5 then 95 of the data falls between 54 2 5 44 and 54 2 5 64 5 P X 41 P P Z 1 21 8869 On the calculator normalcdf 1E99 41 34 5 8 8863 Bottom 4 01 implies find x such that P X x 0401 Look up p 0401 in the body of the table find z 1 75 So x z 34 1 75 5 8 34 10 15 23 85 ounces On the calculator invNorm 0401 34 5 8 23 85 ounces P 30 X 51 P P 1 56 Z 1 13 P Z 1 13 P Z 1 56 8708 0594 8114 On the calculator normalcdf 30 51 42 2 7 8 8115 Top 7 35 implies find x such that P X x 0735 1 0735 9265 look up p 9265 in the body of the table find z 1 45 So x z 42 2 1 45 7 8 42 2 11 31 53 51 years old On the calculator invNorm 9265 42 2 7 8 53 51 years old 9 P X 22 P P Z 2 59 1 P Z 2 59 1 0048 9952 On the calculator normalcdf 22 1E99 42 2 7 8 9952 10 Table find 1 058 for 26 degrees of freedom probability is p 15 On the calculator tcdf 1 058 1E99 26 14989 11 Table we subtract the probability from 1 giving 1 995 005 Find 26 degrees of freedom and then p 005 and the resulting t value is 2 779 On the calculator invT 995 26 2 7787 1 2 3 4 6 7 8