Question

In: Statistics and Probability

The following scenario applies to questions 6-8: A professor reported that final exam scores were left...

The following scenario applies to questions 6-8: A professor reported that final exam scores were left skewed with a mean score of 81.9 and a standard deviation of 6.1.
A student took a random sample of 50 students and calculated the mean score to be 80.3. What is the probability that this student would get a mean of 80.3 or lower?

Suppose the same student took another sample, this time of size 100, and calculated the mean. What is the probability that the student would get a mean of 84 or higher?

Suppose the student is only able to sample 25 students from the class. Can he still calculate the probability of getting an average test score higher than 84? Why or why not?

Solutions

Expert Solution

Solution:

Given: a professor reported that final exam scores were left skewed with a mean score of 81.9 and a standard deviation of 6.1.

Thus Mean= and Standard Deviation =

Part a)

Sample size= n = 50 and sample mean =

We have to find the probability that this student would get a mean of 80.3 or lower .

That is:

Since sample size = n = 50 > 30 , thus we assume large sample and hence using central limit theorem, sampling distribution of sample mean is approximately Normal with mean of sample means =

and standard deviation of sample means is:

Thus find z score:

Thus

Look in z table for z = -1.8 and 0.05 and find corresponding area.

P( Z < -1.85) = 0.0322

Thus

Thus the probability that this student would get a mean of 80.3 or lower is 0.0322

Part b) Sample size = n = 100

Find the probability that the student would get a mean of 84 or higher .

Find z score

where

Thus

Thus we get:

Look in z table for z = 3.4 and 0.04 and find area.

Thus P( Z < 3.4 ) =0.9997

Thus we get:

the probability that the student would get a mean of 84 or higher is 0.0003

Part c) Suppose the student is only able to sample 25 students from the class. Can he still calculate the probability of getting an average test score higher than 84? Why or why not?

Since sample size = n = 25 < 30, thus we can not assume large sample , hence we can not find the probability of getting an average test score higher than 84.


Related Solutions

Student scores on Professor Combs' Stats final exam are normally distributed with a mean of 72...
Student scores on Professor Combs' Stats final exam are normally distributed with a mean of 72 and a standard deviation of 7.5 Find the probability of the following: **(use 4 decimal places)** a.) The probability that one student chosen at random scores above an 77.   b.) The probability that 20 students chosen at random have a mean score above an 77.   c.) The probability that one student chosen at random scores between a 67 and an 77.   d.) The probability...
Student scores on Professor Combs' Stats final exam are normally distributed with a mean of 77...
Student scores on Professor Combs' Stats final exam are normally distributed with a mean of 77 and a standard deviation of 7.5 Find the probability of the following: **(use 4 decimal places)** a.) The probability that one student chosen at random scores above an 82. b.) The probability that 20 students chosen at random have a mean score above an 82. c.) The probability that one student chosen at random scores between a 72 and an 82. d.) The probability...
The final exam scores in a statistics class were normally distributed with a mean of 70...
The final exam scores in a statistics class were normally distributed with a mean of 70 and a standard deviation of five. What is the probability that a student scored less than 55% on the exam?
Exam grades: Scores on a statistics final in a large class were normally distributed with a...
Exam grades: Scores on a statistics final in a large class were normally distributed with a mean of 70 and a standard deviation of 10. Use the TI-84 PLUS calculator to answer the following. Round the answers to at least two decimals. (a) Find the 42nd percentile of the scores. (b) Find the 71st percentile of the scores. (c) The instructor wants to give an A to the students whose scores were in the top 10% of the class. What...
for a final exam in statistics , the mean of the exam scores is u= 75...
for a final exam in statistics , the mean of the exam scores is u= 75 with a standard deviation of o=8. what sample mean marks the top 10% of this distribution for a samples of n= 25 b) what sample means mark the boundaries of the middle 50% for samples of n= 50
"Achievement test scores are declining all around us," brooded Professor Probity. "Here are my final exam...
"Achievement test scores are declining all around us," brooded Professor Probity. "Here are my final exam scores for last year and this year on the same exam." What did Professor Probity discover? (18 pts). This Year | Last Year (this year on the left, last year in the right column) 29    27 27    28 26    25 22    24 19    22 15    20 14    18 10    16 10    12 Indicate: a. a...
An investigator collected data on midterm exam scores and final exam scores of elementary school students;...
An investigator collected data on midterm exam scores and final exam scores of elementary school students; results can summarized as follows. Average SD -------------------------------------------------- Boys' midterm score 70 20 Boys' final score 65 23 girls' midterm score 75 20 girls' final score 80 23 The correlation coefficient between midterm score and final score for the boys was about 0.70; for the girls, it was about the same. If you take the boys and the girls together, the correlation between midterm...
The following are final exam scores for 30 students in an elementary statistics class.
The following are final exam scores for 30 students in an elementary statistics class. 91            59            82            91            79            76            90            69            77            83 59            88            95            72            88            81            77            52            80            96 62            97            76            75            75            89            61            72            90            85 a.              Find the quartiles for this data______________________________. b.             What is the Interquartile Range (IQR)_________________________? c.              What percent of students got at least a 72 on their final exam______________? d.              Build a boxplot using the graphing calculator.
Suppose a professor gives an exam to a class of 40 students and the scores are...
Suppose a professor gives an exam to a class of 40 students and the scores are as follows. (Type the data set in StatCrunch.) 35 44 46 47 47 48 49 51 53 54 55 55 57 57 57 58 59 59 59 59 60 60 60 60 60 62 62 62 64 68 69 70 72 73 73 75 75 77 82 88 Top 10% of scores receive an A Bottom 10% of scores receive an F Scores between...
A statistics professor claims that the average score on the Final Exam was 83. A group...
A statistics professor claims that the average score on the Final Exam was 83. A group of students believes that the average grade was lower than that. They wish to test the professor's claim at the  α=0.05α=0.05 level of significance. (Round your results to three decimal places) Which would be correct hypotheses for this test? H0:μ=83H0:μ=83, H1:μ≠83H1:μ≠83 H0:μ≠83H0:μ≠83, H1:μ=83H1:μ=83 H0:μ=83H0:μ=83, H1:μ>83H1:μ>83 H0:μ=83H0:μ=83, H1:μ<83H1:μ<83 H0:μ<83H0:μ<83, H1:μ=83H1:μ=83 A random sample of statistics students had the Final Exam scores shown below. Assuming that the...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT