Question

1. Probabilities with z-scores and standardized scores

Remember that you have been given the mean (μ = 70) and population standard deviation (σ = 8) for the raw scores from this verbal intelligence test and that the standardized scores have a mean of 100 and standard deviation of 25.

1. What % of the distribution is expected to fall below the raw score of 80?

_________________

2. What % of the distribution is expected to fall below a standardized score of 60?

_________________

3. You test another English class and obtain a mean raw score of 72 (n = 100). What is the probability of selecting a random sample of n = 100 scores with a sample mean this large or larger?

_________________

4. Another school psychologist informs you that she obtained a standardized score sample mean of 95 for the senior high school students enrolled in English (n = 49). What is the probability of selecting a random sample of n = 49 scores with a sample mean this larger or larger?

_________________

Answer 1

1)

for normal distribution z score =(X-μ)/σ
here mean=       μ=	70
std deviation   =σ=	8.0000

% of the distribution is expected to fall below the raw score of 80 :

probability =

P(X<80)

=

P(Z<1.25)=

89.44%

2)

% of the distribution is expected to fall below a standardized score of 60

probability =

P(X<60)

=

P(Z<-1.6)=

5.48%

3)

sample size       =n=	100
std error=σx̅=σ/√n=	0.8000

probability =

P(X>72)

=

P(Z>2.5)=

1-P(Z<2.5)=

1-0.9938=

0.0062

4)

sample size       =n=	49
std error=σx̅=σ/√n=	3.5714

probability =

P(X>95)

=

P(Z>-1.4)=

1-P(Z<-1.4)=

1-0.0808=

0.9192