Question

The college Physical Education Department offered an Advanced First Aid course last summer. The scores on the comprehensive final exam were normally distributed, and the z scores for some of the students are shown below.

Robert, 1.00	Juan, 1.67	Susan, –2.16
Joel, 0.00	Jan, –0.71	Linda, 1.68

(a) Which of these students scored above the mean? (Select all that apply.)

Robert
Joel
Jan
Juan
Susan
Linda

(b) Which of these students scored on the mean? (Select all that apply.)

Robert
Joel
Jan
Juan
Susan
Linda

(c) Which of these students scored below the mean? (Select all that apply.)

Robert
Joel
Jan
Juan
Susan
Linda

(d) If the mean score was μ = 154 with standard deviation σ = 24, what was the final exam score for each student? (Round your answers to the nearest whole number.)

Robert
Joel
Jan
Juan
Susan
Linda

Answer 1

Here, we are given the following z scores of the marks of students which are Normally distributed.

Robert :- 1.00     Juan:- 1.67     Susan:- –2.16

Joel:- 0.00     Jan:- –0.71     Linda:- 1.68.

Using this information, we have to answer the following questions:-

(a) Which of these students scored above the mean?

ANSWER:- The Z , i.e. Standard Normal Variable has

N(0,1). i.e. Mean= 0, Variance= 1

So, the students who scored above the mean score,

i.e. above Z=0.

Hence, Students for which Z>0 are Robert, Juan and Linda.

Hence, Robert, Juan and Linda scored above the mean score.

(b) Which of these students scored on the mean?

ANSWER:- The Z , i.e. Standard Normal Variable has

N(0,1). i.e. Mean= 0, Variance= 1

So, the students who scored on the mean score,

i.e. have Z=0.

Hence, Students for which Z=0 is Joel.

Hence, Joel has scored on the mean score.

(c) Which of these students scored below the mean?

ANSWER:- The Z , i.e. Standard Normal Variable has

N(0,1). i.e. Mean= 0, Variance= 1

So, the students who scored below the mean score,

i.e. below Z=0.

Hence, Students for which Z<0 are Jan and Susan.

Hence, Jan and Susan scored below the mean score.

(d) If the mean score was μ = 154 with standard deviation σ = 24, what was the final score for each student? (Round your answers to the nearest whole number.)

ANSWER:-

We know that ,

Now, rearranging the above Equation, we get:-

Now, we are given that,

μ = 154 with standard deviation σ = 24,

Hence, the value of X , the final score for each of the following is:-

Robert :- Z= 1.00 then

X= 154+(1)*(24) = 178

Joel:- Z= 0.00 then

X= 154+(0)*(24) = 154

Jan:- Z= –0.71 then

X= 154+(-0.71)*(24) = 136.96~ 137(approximately)

Juan:- Z= 1.67 then

X= 154+(1.67)*(24)= 194.08 ~194(approximately)

Susan:- Z= –2.1 then

X=154+(-2.1)*(24)= 103.6~104(approximately)

Linda:- Z= 1.68 then

X=154+(1.68)*(24)= 194.32~194(approximately)

These are the final scores of each student.

This answers your question.

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