In: Statistics and Probability
Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us.
(a)
Suppose n = 31 and p = 0.17. (For each answer,
enter a number. Use 2 decimal places.)
n·p =
n·q =
Can we approximate p̂ by a normal distribution? Why? (Fill
in the blank. There are four answer blanks. A blank is represented
by _____.)
_____, p̂ _____ be approximated by a normal random
variable because _____ _____.
first blank
YesNo
second blank
cancannot
third blank
both n·p and n·q exceed n·p does not exceed n·q exceeds n·p and n·q do not exceed n·q does not exceed n·p exceeds
fourth blank (Enter an exact number.)
What are the values of μp̂ and
σp̂? (For each answer, enter a number.
Use 3 decimal places.)
μp̂ = mu sub p hat =
σp̂ = sigma sub p hat =
(b)
Suppose n = 25 and p = 0.15. Can we
safely approximate p̂ by a normal distribution? Why or why
not? (Fill in the blank. There are four answer blanks. A blank is
represented by _____.)
_____, p̂ _____ be approximated by a normal random
variable because _____ _____.
first blank
YesNo
second blank
cancannot
third blank
both n·p and n·q exceed n·p does not exceed n·q exceeds n·p and n·q do not exceed n·q does not exceed n·p exceeds
fourth blank (Enter an exact number.)
(c)
Suppose n = 55 and p = 0.29. (For each
answer, enter a number. Use 2 decimal places.)
n·p =
n·q =
Can we approximate p̂ by a normal distribution? Why? (Fill
in the blank. There are four answer blanks. A blank is represented
by _____.)
_____, p̂ _____ be approximated by a normal random
variable because _____ _____.
first blank
YesNo
second blank
cancannot
third blank
both n·p and n·q exceed n·p does not exceed n·q exceeds n·p and n·q do not exceed n·q does not exceed n·p exceeds
fourth blank (Enter an exact number.)
What are the values of μp̂ and
σp̂? (For each answer, enter a number.
Use 3 decimal places.)
μp̂ = mu sub p hat =
σp̂ = sigma sub p hat =
solution:
a) Given data for Binomial distribution
n = 31 , p = 0.17 then q = 1- p = 1 - 0.17 = 0.83
Here , np = 31*0.17 = 5.27
nq = 31*0.83 = 25.73
To approximate by a normal distribution
then np>5 and nq>5 (satisfies only when n is large and p is close to 0.5)
Here, observe that both np>5 and nq>5
Answer: Yes,can be approximated by a normal random variable because both np and nq exceed
Here, = np = 31*0.17 = 5.27
= = = 2.091
b) Given data for Binomial distribution
n = 25 , p = 0.15 then q = 1- p = 1 - 0.15 = 0.85
Here , np = 25*0.15 = 3.75
nq = 25*0.85 = 21.25
To approximate by a normal distribution
then np>5 and nq>5 (satisfies only when n is large and p is close to 0.5)
Here, observe that both np<5 and nq>5
Answer: No,cannot be approximated by a normal random variable because np doesn't exceed and nq exceed
c) Given data for Binomial distribution
n = 55 , p = 0.29 then q = 1- p = 1 - 0.29 = 0.71
Here , np = 55*0.29 = 15.95
nq = 55*0.71 = 39.05
To approximate by a normal distribution
then np>5 and nq>5 (satisfies only when n is large and p is close to 0.5)
Here, observe that both np>5 and nq>5
Answer: Yes,can be approximated by a normal random variable because both np and nq exceed
Here, = np = 55*0.29 = 15.95
= = = 3.365