Question

a. Major League Baseball now records information about every pitch thrown in every game of every season. Statistician Jim Albert compiled data about every pitch thrown by 20 starting pitchers during the 2009 MLB season. The data set included the type of pitch thrown (curveball, changeup, slider, etc.) as well as the speed of the ball as it left the pitcher’s hand. A histogram of speeds for all 30,740 four-seam fastballs thrown by these pitchers during the 2009 season is shown below, from which we can see that the speeds of these fastballs follow a Normal model with mean μ = 92.12 mph and a standard deviation of σ = 2.43 mph.

b. Scores for a common standardized college aptitude test are normally distributed with a mean of 480 and a standard deviation of 105. Randomly selected men are given a Test Preparation Course before taking this test. Assume, for sake of argument, that the preparation course has no effect.

If 1 of the men is randomly selected, find the probability that his score is at least 532.5.
P(X > 532.5) =
Enter your answer as a number accurate to 4 decimal places.

If 9 of the men are randomly selected, find the probability that their mean score is at least 532.5.
P(M > 532.5) =
Enter your answer as a number accurate to 4 decimal places.

c. A population of values has a normal distribution with μ=63.8μ=63.8 and σ=88.3σ=88.3. You intend to draw a random sample of size n=98n=98.

Find the probability that a single randomly selected value is less than 83.4.
P(X < 83.4) =

Find the probability that a sample of size n=98n=98 is randomly selected with a mean less than 83.4.
P(M < 83.4) =

Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

d. Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find the probability of obtaining a reading greater than 3.186°C. Round answers to 4 decimal places.

P(Z>3.186)=P(Z>3.186)=

e.

Assume that the readings at freezing on a bundle of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find the probability of obtaining a reading between -0.398°C and 1.554°C.

P(−0.398<Z<1.554)=P(-0.398<Z<1.554)=

Compute the z-score of pitch with speed 84.9 mph. (Round your answer to two decimal places.)



Approximately what fraction of these four-seam fastballs would you expect to have speeds between 88 mph and 93 mph? (Express your answer as a decimal, not a percent, and round to three decimal places.)



Approximately what fraction of these four-seam fastballs would you expect to have speeds below 88 mph? (Express your answer as a decimal, not a percent, and round to three decimal places.)



A baseball fan wishes to identify the four-seam fastballs among the slowest 4% of all such pitches. Below what speed must a four-seam fastball be in order to be included in the slowest 4%? (Round your answer to the nearest 0.1 mph.)

mph

Answer 1

b)

µ =    480                  
σ =    105                  
                      
P ( X ≥   532.50   ) = P( (X-µ)/σ ≥ (532.5-480) / 105)              
= P(Z ≥   0.500   ) = P( Z <   -0.500   ) =    0.3085   (answer)

µ =    480              
σ = 105/ sqrt(9) = 35              
                  
P ( X ≥   532.50   ) = P( (X-µ)/σ ≥ (532.5-480) / 35)          
= P(Z ≥   1.500   ) = P( Z <   -1.500   ) =    0.0668

c)

µ =    63.8      
σ =    88.3      
          
P( X ≤    83.4   ) = P( (X-µ)/σ ≤ (83.4-63.8) /88.3)  
=P(Z ≤   0.222   ) =   0.5878

µ =    63.8      
σ = 88.3/sqrt(98) = 8.9196
          
P( X ≤    83.4   ) = P( (X-µ)/σ ≤ (83.4-63.8) /8.91964696839602)  
=P(Z ≤   2.197   ) =   0.9860

d)

µ =    0              
σ =    1              
                  
P ( X ≥   3.19   ) = P( (X-µ)/σ ≥ (3.186-0) / 1)          
= P(Z ≥   3.186   ) = P( Z <   -3.186   ) =    0.0007

Thanks in advance!

revert back for doubt

Please upvote

e)

µ =    0                              
σ =    1                              
we need to calculate probability for ,                                  
P (   -0.398   < X <   1.554   )                  
=P( (-0.398-0)/1 < (X-µ)/σ < (1.554-0)/1 )                                  
                                  
P (    -0.398   < Z <    1.554   )                   
= P ( Z <    1.554   ) - P ( Z <   -0.398   ) =    0.9399   -    0.3453   =    0.5946