1. which statement is true of normally distributed data?

a. approximately 68% of data falls within 2 standard deviations (±2) of the mean.
b. approximately 68% of data falls within 1 standard deviation (±1) of the mean.
c. approximately 75% of data falls within 2 standard deviations (±2) of the mean.
d. approximately 75% of data falls within 1 standard deviation (±1) of the mean.

2. which statement is true of normally distributed data?

a. approximately 90% of data falls within 2 standard deviations (±2) of the mean.
b. approximately 90% of data falls within 3 standard deviations of (±3) the mean.
c. approximately 95% of data falls within 2 standard deviations (±2) of the mean.
d. approximately 95% of data falls within 3 standard deviations (±3) of the mean.

3. which statement describes data that is normally distributed?

a. the majority of the data is to the left of the mean.
b. the majority of the data is to the right of the mean.
c. the majority of the data is in the center with the mean.
d. the majority of the data is split to the right and left of the mean.

4. for the following data set, calculate the percentage of data points that fall within one standard deviation of the mean, and compare the result to the expected percentage of a normal distribution.
{55, 54, 66, 38, 53, 56, 57, 66, 45, 65}

a. 50%; this percentage is lower than expected in a normal distribution.
b. 60%; this percentage is lower than expected in a normal distribution.
c. 70%; this percentage is close to the expected percentage in a normal distribution.
d. 80%; this percentage is higher than expected in a normal distribution.

5. for the following data set, calculate the percentage of data points that fall within one standard deviation of the mean, and compare the result to the expected percentage of a normal distribution.
{50, 46, 54, 51, 29, 52, 48, 54, 47, 48}

a. 80%; this percentage is lower than expected in a normal distribution.
b. 80%; this percentage is higher than expected in a normal distribution.
c. 90%; this percentage is lower than expected in a normal distribution.
d. 90%; this percentage is higher than expected in a normal distribution.

6. based on the 2017 season, the houston astros have a winning percentage of .623. use the binomial model to find the probability that the astros will win 4 of their next 6 games.

p(x)={n/x! (n-x)}p^xq^n-x

a. 12.4%
b. 24.7%
c. 32.1%
d. 62.3%

7. a student randomly guesses on 10 true or false questions. use the binomial model to find the probability that the student gets 7 out of the 10 questions right.

p(x)={n/x! (n-x)}p^xq^n-x

a. 11.8%
b. 20.9%
c. 32.4%
d. 50%

8. 5% of batteries produced at a factory are defective. use the binomial model to find the probability that 1 battery in a pack of 16 is defective.

p(x)={n/x! (n-x)}p^xq^n-x

a. 5%
b. 15%
c. 23%
d. 37%

9. in the u. s., 51% of all births are boys, and 49% are girls. use the binomial model to find the probability that 6 out of 10 births at a local hospital are boys.

p(x)={n/x! (n-x)}p^xq^n-x

a. 17%
b. 21%
c. 49%
d. 51%

10. a target is made of a yellow square inside of a green square. what is the theoretical probability that a dart will hit the green square if the side length of the yellow square is 4, and the side length of the green square is 8?

a. 0.25
b. 0.5
c. 0.75
d. 0.8

answer:

1. b

2. c

3. c

4. b

5. d

6. c

7. a

8. d

9. b

10. c

step-by-step explanation:

the standard deviation tells us the "dispersion" there is from the mean (average).

for normal distributions,

we know that 68% of the data lies within 1 standard deviation, 95% of the data lies within 2 standard deviation, and 99% of the data lies within 3 standard deviations of the mean (average).

** they all lie to the left (negative) and right (positive) of the mean

1. from the discussion above, we can say that "approximately 68% of data falls within 1 standard deviation (±1) of the mean." -- answer choice b is right.

2. also from above discussion, we can say that "approximately 95% of data falls within 2 standard deviations (±2) of the mean." -- answer choice c is right.

3. a normal distribution takes the shape of a "bell curve" where most of the data is in the center (with the mean) and gradually decreases tot he left and right symmetrically. answer choice c is right.

4. the mean of the data set is  

the standard deviation is given by the formula:

this means that we take each value (from the data set) and subtract the mean from it and take square of it. we sum all of them and divide by the number of numbers (n=10). then we square root the number to get standard deviation.

** being such a big calculation, i am not showing this, rather writing out the answer using a calculator ***

standard deviation = 9.06

the interval within 1 standard deviation of the mean is:

(55.5 - 9.06 , 55.5 + 9.06) = (46.44, 64.56)

the number of data that falls within this interval is 6, or 60% (6 out of 10)

as we know from before, 68% should fall within 1 st. dev. of the mean, but this number is lower.

answer choice b is right.

5. we follow the same process as question #4.

mean =

the standard deviation is given by the formula:

** being such a big calculation, i am not showing this, rather writing out the answer using a calculator ***

standard deviation = 7.20

the interval within 1 standard deviation of the mean is:

(47.9 - 7.20 , 47.9 + 7.20) = (40.7, 55.1)

the number of data that falls within this interval is 9, or 90% (9 out of 10)

as we know from before, 68% should fall within 1 st. dev. of the mean, but this number is much higher.

answer choice d is right.

6. the formula is given as:

where

n is the total games (n=6)

x is what we are looking for in the question (x=4 since we want probability of 4 win)

p is probability of win (given as 0.623)

q is probability of losing (q = 1 - p = 1 - 0.623 = 0.377)

substituting all the information in the formula we get:

which, in percentage, 32.1%

7.

this is similar to question #6.

the formula is:

where

n is the total number of questions (n=10)

x is what we are looking for in the question (x=7 since we want probability of 7 right)

p is probability of right (since true or false, probability is 0.5)

q is probability of losing (q = 1 - p = 1 - 0.5 = 0.5)

substituting all the information in the formula we get:

in percentage, that's 11.8%

8.

the formula is:

where

n is the total number of batteries (n=16)

x is what we are looking for in the question (x=1 since we want probability of 1 of them being defective)

p is probability of defective (given as 0.05)

q is probability of not defective (q = 1 - p = 1 - 0.05 = 0.95)

substituting all the information in the formula we get:

in percentage, that's 37%

9.

the formula is:

where

n is the total number of births (n=10)

x is what we are looking for in the question (x=6 since we want probability of 6 of them being boys)

p is probability of boy (given as 0.51)

q is probability of not boy (q = 1 - p = 1 - 0.51 = 0.49)

substituting all the information in the formula we get:

in percentage, that's 21%

10. based on the target description, the green region is:

green region = area of green - area of yellow

since area of square is side * side , so we have

area of green = 8*8=64

area of yellow = 4*4=16

green region = 64 - 16 = 48

total area is the area of the green square, which is 64.

since we need probability of hitting green region, we divide 48 (area of green region) by 64 (total area of the target). hence we have:

probability of hitting =

answer:

step-by-step explanation:

1) b. approximately 68% of data falls within 1 standard deviation (±1) of the mean.

2) c. approximately 95% of data falls within 2 standard deviations (±2) of the mean.

3) d. the majority of the data is split to the right and left of the mean.

4) mean =55.5 and sds = 9.6: std error = 3.2

i.e. (55.5-9.6. 55.5+9.6)

actual data lying is 70%

d. 80%; this percentage is higher than expected in a normal distribution.

5) mean = 47.5: sd= 7.20

within 1 std dev= (40.3,52.7)

b. 80%; this percentage is higher than expected in a normal distribution.

6) c. 32.1%

7) a. 11.8%

8) d. 37%

9)b. 21%

10) area of yellow square/area of green square = 16/64 = 0.25

a. 0.25

answer: c is correct, a regular pentagon