Question

1. The upper leg length of 20 to 29 year olds males is normally distributed with mean length of 43.7 cm and a standard deviation of 4.2 cm
   1. What is the probability that a randomly selected 20-29 year old male has an upper leg length of that is less than 40 cm?
   2. Suppose a random sample of 9 males who are 20-29 years old is obtained. What is the probability that their mean leg length is less than 40 cm?
   3. A random sample of 15 males who are 20-29 years old results in a mean leg length of 46 cm or more. Do you find this result unusual? Why?

2. The reading rates of second grade students is approximately normal with a mean of 90 words per minute and a standard deviation of 10 words per minute
   1. What is the probability that a random student will read more than 95 words per minute?
   2. What is the probability that a random sample of 24 second grade students will result a mean reading rate of more than 95 words per minute?
   3. A teacher initiated a new reading program at school. After 10 weeks on the program, it was found that the mean reading speed of a random sample of 20 second grade students was 93 words per minute? Is this result unusual? What does this suggest about the effectiveness of the program?

Answer 1

1)a)

for normal distribution z score =(X-μ)/σ
here mean=       μ=	43.7
std deviation   =σ=	4.2000

probability =

P(X<40)

=

P(Z<-0.88)=

0.1894

b)

sample size       =n=	9
std error=σx̅=σ/√n=	1.4000

probability =

P(X<40)

=

P(Z<-2.64)=

0.0041

c)

sample size       =n=	15
std error=σx̅=σ/√n=	1.0844

probability =

P(X>46)

=

P(Z>2.12)=

1-P(Z<2.12)=

1-0.9830=

0.0170

2)

for normal distribution z score =(X-μ)/σ
here mean=       μ=	90
std deviation   =σ=	10.0000

probability =

P(X>95)

=

P(Z>0.5)=

1-P(Z<0.5)=

1-0.6915=

0.3085

b)

sample size       =n=	24
std error=σx̅=σ/√n=	2.0412

probability =

P(X>95)

=

P(Z>2.45)=

1-P(Z<2.45)=

1-0.9929=

0.0071

c)

sample size       =n=	20
std error=σx̅=σ/√n=	2.2361

probability =

P(X>93)

=

P(Z>1.34)=

1-P(Z<1.34)=

1-0.9099=

0.0901

as probability of that happening is not less than 0.05 level ; therefore it is not unusual.