0503HW_sol  5.3 Binomial

Link: http://www.rmower.com/statistics/stat_lecture/0065_discrete_rand_var_all.pdf

Answers: 1) Procedure results in a binomial distribution.  2) 0.117   3) 0.184   4) 0.119   5) 3.4   6) 7.3   7) Yes

Solutions below:

1)  (Original Problem) Determine whether the given procedure results in a binomial distribution. If not, state the reason why. Rolling a single die 47 times, keeping track of the "fives" rolled.

 

    Solution:  To be a binomial, it must satisfy 4 conditions. Does the above satisfy

    1) The experiment is made up of n identical trials (or steps).  Yes. There are 47 trials.   
    2) Each of the n trials has only two possible outcomes. Yes. It's a 5 ir it's not a 5.   
    3) The probabilities of the two outcomes of the trials remain constant. Yes. When rolling a die, the probability of the outcomes remain constant.
    4) Each trial is independent of the rest. Yes. What you get each time has no influence on wht you get the next time.

2) (Original Problem) Assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial. Round to three decimal places.  n = 10, x = 7 , p = 0.5

    Solution1: On the TI-30 XS, the TI-83/84 Plus:
Use the formula .  Enter a 10 then press MATH then PRB then select nCr then enter a 7 then enter a x (times *) then enter .5^7x.5^3 then press enter.  (Note: To access nCr on the TI-30 X S you don't have to press MATH...just press PRB.) It should look like this: "10 nCr 6x.5^7x.5^3" (Note: On some models/operating systems, it might look like this: "10 nCr 6x.57x.53"). Note: "x" means a times sign; usually an asterisk "
*".

    Solution2: On the TI-83/84 Plus built-in function "binomialpdf(:  Press 2nd DISTR then select binomialpdf( and enter the arguments as follows: binomialpdf(10,.5,{7}). [Make sure you use the correct { or } where needed...the calculator knows the difference between a ")" and a "}".  Note: The arguments are binomialpdf(n,p,{x1,x2,x3,...xi}).

See: http://www.rmower.com/statistics/calculator_built-in_handouts/calculator_binomial_probabilities.PDF

    Solution3: On the TI-83/84 Plus using the program "BINOMIAL":  Press prgm and select BINOMIAL and press ENTER to run the program.  Follow the prompts.  The program is in Group CH0104Vn and must have been downloaded (by your instructor).  Ask in class.


3) (Original Problem) Find the indicated probability. A tennis player makes a successful first serve 48% of the time. If she serves 9 times, what is the probability that she gets exactly 3 first serves in? Assume that each serve is independent of the others.


    Solution1: On the TI-30 XS, the TI-83/84 Plus: Use the formula .  Enter a 9 then press MATH then PRB then select nCr then enter a 3 then enter a x (times *) then enter .48^3x.52^6 then press enter.  (Note: To access nCr on the TI-30 X S you don't have to press MATH...just press PRB.) It should look like this: "9 nCr 3x.48^3x.52^6" (Note: On some models/operating systems, it might look like this: "9 nCr 3x.483x.526"). Note: "x" means a times sign; usually an asterisk "*".

    Solution2: On the TI-83/84 Plus built-in function "binomialpdf(:  Press 2nd DISTR then select binomialpdf( and enter the arguments as follows: binomialpdf(9,.48,{3}). [Make sure you use the correct { or } where needed...the calculator knows the difference between a ")" and a "}".  Note: The arguments are binomialpdf(n,p,{x1,x2,x3,...xi}).

See: http://www.rmower.com/statistics/calculator_built-in_handouts/calculator_binomial_probabilities.PDF

    Solution3: On the TI-83/84 Plus using the program "BINOMIAL":  Press prgm and select BINOMIAL and press ENTER to run the program.  Follow the prompts.  The program is in Group CH0104Vn and must have been downloaded (by your instructor).  Ask in class.

 

4) (Original Problem) Find the indicated probability. Round to three decimal places.  In a study, 43% of adults questioned reported that their health was excellent. A researcher wishes to study the health of people living close to a nuclear power plant. Among 13 adults randomly selected from this area, only 3 reported that their health was excellent. Find the probability that when 13 adults are randomly selected, 3 or fewer are in excellent health.

 

    Solution1: On the TI-30 XS, the TI-83/84 Plus: Use the formula You must do P(x 3) = P(0) + P(1) + P(2) + P(3).  Enter a 13 then press MATH then PRB then select nCr then enter a 0 then enter a x (times *) then enter .43^0x.57^13 then + then enter a 13 then press MATH then PRB then select nCr then enter a 1 then enter a x (times *) then enter .43^1x.57^12 then + then enter a 13 then press MATH then PRB then select nCr then enter a 2 then enter a x (times *) then enter .43^2x.57^11 then + then enter a 13 then press MATH then PRB then select nCr then enter a 3 then enter a x (times *) then enter .43^3x.57^10 then press enter.  (Note: To access nCr on the TI-30 X S you don't have to press MATH...just press PRB.) It should look like this: "13 nCr 0x.43^0x.57^13 + 13 nCr 1x.43^1x.57^12 + 13 nCr 2x.43^2x.57^11 + 13 nCr 3x.43^3x.57^10" (Note: On some models/operating systems, it might have raised exponents like 13 nCr 0x.430x.5713). Note: "x" means a times sign; usually an asterisk "*".

    Solution2: On the TI-83/84 Plus built-in function "binomialpdf(:  You must do P(x 3) = P(0) + P(1) + P(2) + P(3).  Press 2nd DISTR then select binomialpdf( and enter the arguments as follows: sum(binomialpdf(13,.43,{0,1,2,3})). [Make sure you use the correct { or } where needed...the calculator knows the difference between a ")" and a "}".  Note: The arguments are binomialpdf(n,p,{x1,x2,x3,...xi}).  You can leave closing parenthesis off like "sum(binomialpdf(13,.43,{0,1,2,3".  To find the "sum(" command, press 2nd then List then select MATH then sum(.

See: http://www.rmower.com/statistics/calculator_built-in_handouts/calculator_binomial_probabilities.PDF

    Solution3: On the TI-83/84 Plus using the program "BINOMIAL":  You must do P(x 3) = P(0) + P(1) + P(2) + P(3).  Press prgm and select BINOMIAL and press ENTER to run the program.  Follow the prompts.  The program is in Group CH0104Vn and must have been downloaded (by your instructor).  Ask in class.

    Using Excel - - =BINOMDIST(A2,13,0.43,FALSE)

x P(x)
0 0.00067
1 0.006575
2 0.029762
3 0.082323
Total 0.11933

 

5) (Original Problem) The probability is 0.2 that a person shopping at a certain store will spend less than $20. For groups of size 17, find the mean number who spend less than $20.

 

    Solution: Use = np.  = 17(.2) = 3.4.

 

6) (Original Problem) In a certain town, 42% of voters favor a given ballot measure. For groups of 30 voters, find the variance for the number who favor the measure.

 

    Solution: 

 

7) (Original Problem)  Determine if the outcome is unusual. Consider as unusual any result that differs from the mean by more than 2 standard deviations. That is, unusual values are either less than μ - 2σ or greater than μ + 2σ.  According to AccuData Media Research, 36% of televisions within the Chicago city limits are tuned to "Eyewitness News" at 5:00 pm on Sunday nights. At 5:00 pm on a given Sunday, 2500 such televisions are randomly selected and checked to determine what is being watched. Would it be unusual to find that 954 of the 2500 televisions are tuned to "Eyewitness News"?

Solution:  Find the mean and standard deviation.   = np.  = 2500(.36) = 900.  .  Now find the usual number limits.    - 2 s = 900 - 2 x 24 = 852 and + 2 s = 900 + 2 x 24 = 948.  The number, 954 is not between the usual number limits of 852 and 948 and hence, is unusual.