Question

In: Statistics and Probability

Suppose we have a binomial experiment in which success is defined to be a particular quality...

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 by a normal distribution? Why? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, _____ 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 μ and σ? (For each answer, enter a number. Use 3 decimal places.)
μ = mu sub p hat =

σ = sigma sub p hat =

(b)

Suppose n = 25 and​​​​​​​ p = 0.15. Can we safely approximate by a normal distribution? Why or why not? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, _____ 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 by a normal distribution? Why? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, _____ 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 μ and σ? (For each answer, enter a number. Use 3 decimal places.)
μ = mu sub p hat =

σ = sigma sub p hat =

Solutions

Expert Solution

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


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