## binomial theorem

Algebra 2-Assignment 11. ) Use the binomial theorem to write the binomial expansion of (x3y

2 ) 5 .

2.) Use the binomial theorem to find the 12th term in the binomial expansion of (2

vx+½ y) 19

3.) Find the 8th number in the 11th row (n=11) of Pascal’s triangle.

4.) Show that the sum of the 34th and 35th numbers in the 117th row of Pascal’s

triangle is equal to the 85th number in the 118th row of the triangle.

5.) Over 350 students took a college calculus final exam. The scores of the students

follow a normal distribution. Using the information given below, determine the mean and

the standard deviation for the students’ scores. Give your answers to the nearest tenth

of a percent.

? 32% of the students scored either less that 63.3% or more than 85.9%

? Only the top 2.5% of students scored higher than a 97% on the exam

Mean: _____________________ Standard Deviation: ___________________

6.) Alexander, a meteorologist, is studying the daily high temperatures over the past 50

years during the month of July for the cities of Chicago and Phoenix (a sample size of

1550 July days for each city). Alexander found that in Chicago, the middle 95% of July

days have a high temperature between 70 °F and 98°F, while in Phoenix the middle 95% of

July days have a high temperature between 94°F and 110°F. On one particular July day, the

temperature in Phoenix is exactly 17°F higher than the temperature in Chicago. Which of the

following is not a valid pair of zscores

for the two cities? Circle your answer.

(A)Chicago: z=1.7143 (B)Chicago: z=0.8571

Phoenix: z=3.2500

Phoenix: z=1.2500

(C)Chicago: z=0.4286 (D)Chicago: z=1.5714

Phoenix: z=0.7500 Phoenix: z=2.5000

7.) A polling company defines a “likely voter” as someone who has voted in each of the previous

two elections. The company asks 815 likely voters which candidate they support in the

upcoming election. FInd the margin of error for the poll to the nearest tenth of a percent.

8.) On a fair coin, each side has an equal probability of coming up. When a fair coin is flipped n

times, the most likely outcome (the mean) is that each side will come up n /2 times, with a

standard deviation of o= v n /2. Megan flips a coin 50 times. She will conclude that the coin

is unfair if the number of times either side is flipped is outside 1.5 standard deviations of

the mean. Show your calculations to complete the statement below.

The coin will be considered fair if her results show “heads” at least ________ times and

no more than _________ times.

9.) A studio needs 3 actors for their short comedy film. The 6 actors available are Moe,

Larry, Curly, Shemp, Joe, and Curly Joe.

a.) How many different combinations of 3 actors can the studio choose?

b.) Suppose Moe must be one of the actors. How many different combinations of 3

actors can the studio now choose?

c.) Suppose Moe must be one of the actors, but also that Shemp refuses to work with

either Joe or Curly Joe. Now how many different combinations of 3 actors can the studio

select?

10.) Compared to the 25 winter seasons from 19812006,

the 35.6 inches of snowfall in

Chicago during the 20062007

season was snowier than 53.98% of them and the 60.3

inches in the 20072008

season was more snowfall than 99.65% of them.

a.) Calculate the amount of the snowfall needed in the 20082009

season to maintain

the average established from 19812006

seasons.

b.) Chicago got 52.7 inches of snowfall in the 20082009

winter season. Calculate the

percentile to accurately complete the statement below.

The 20082009

winter saw more snowfall than ______% of winters from 19812006.

c.) Find the average seasonal snowfall from 1981 to 2009 to the nearest tenth of an

inch.

11.) The tables below give the ten most popular names given to girls born in the U.S. in

the years 1974 and 1999 and the percentages of girls given those names. Consider a

group of 20 girls born in 1974, and another group of 20 born in 1999. Use the data in

the tables to do problems 1213

regarding these groups.

1974:

#1 Jennifer 4.03% #6 Kimberly 1.43%

#2 Amy 1.89% #7 Melissa 1.42%

#3 Michelle 1.65% #8 Lisa 1.26%

#4 Heather 1.48% #9 Stephanie 1.08%

#5 Angela 1.46% #10 Rebecca 0.97%

1999:

#1 Emily 1.36% #6 Ashley 0.93%

#2 Hannah 1.11% #7 Madison 0.93%

#3 Alexis 0.99% #8 Taylor 0.87%

#4 Sarah 0.98% #9 Jessica 0.84%

#5 Samantha 0.98% #10 Elizabeth 0.79%

12.) The name Jennifer was approximately 3 times as popular in 1974 as Emily was in

1999, despite each being the #1 name in their respective years. Determine by

calculation if you are 3 times as likely to have more than one Jennifer in the 1974 group

of 20 as you are to have multiple Emilys in the 1999 group of 20.

13.) A sample of 5 girls is chosen at random from each group. Complete the probability

distribution table for the number of girls in each sample with a top ten first name. Give

your probabilities to 4 decimal places.

X 0 1 2 3 4 5

1974

P(X)

1999

P(X)

14.) Draw a histogram of the binomial distribution for each sample. Use open boxes for

the 1974 sample and shaded boxes for the 1999 sample.

15.) Use your histogram to fill in the blanks below with the most appropriate values. In a

group of 5 girls born in 1999, it is most likely that ____ of them will have a top ten name;

for 5 girls born in 1974 it is most likely that 1 or ____ of them will have a top ten name.

16.) A large college class has 54 women and 42 men as students. Ten of the students

are chosen at random.

a.) What is the probability that the chosen group of 10 will include 5 women and 5 men?

Give your answer to the 4 decimal places.

b.) List all possible makeups of the group that have a probability of less that 1%. Give

your probability to 4 decimal places.

c.) Is the probability distribution symmetric or skewed? Justify your answer.