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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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I did this question fully but only got the correct answer for 1 a). For b) I didnt get it correct can you locate where I made my mistake for it than redo it explaining everything in it

I will paste the picture of full solutions for a) and b) but only help me with b)

My question and answers are posted first and the correct solution are in the pictures (only help for (b))

In a data management exam, the scores of 567 participants followed a continuous normal distribution. It is known that the mean from all of the students is 75, and to reach the top 1%, students need to score more than (exactly) 92.

a) Estimate the number of students who scored between 60 and 70.

Using standard deviation

P(X<=92) = P(Z<=X-mean/SD) = P(Z<=92-75/SD) = P(Z<=17/SD) = 0.99 where Z follows standard normal distribution with mean 0 and SD (standard deviation) 1.

Using standard normal distribution

17/SD = 2.326

SD (standard deviation) = 17/2.326 = 7.31

Probability that student scores b/w 60 and 70 = P(60<X<70)

Using standardization: P(60<X<70) = P(60-75/7.31 < Z < 70 -75/7.31) = P(-15/7.31 < Z -5/7.31) = 0.2268

Number of students scoring b/w 60 and 70 = 567*0.2268 = 128.5956 = 129

b) 10 students are good friends. After knowing each other’s marks, they came up with a 95% confidence interval of 65 ± 12 for the population mean. Is their estimation of the population mean still correct, if they used a 90% confidence level?

Given 95% CI is 65±12 = (53, 77)

It means that we are 95% (more than 90%) confident that the true mean score lies b/w 53 and 77.

And the true mean score i.e., 75 lies in the given confident interval.

we cannot conclude that these students performed worse than average with 90% confidence.

5) In a data management exam, the scores of 567 students followed a continuous normal distribution.
It is known that the mean of the exam is 75, and to reach the top 1%, students need to score 92 or
above.
a) Estimate the number of students who scored between 60 and 70.
Top 1 percent means 99 percentile. z-score = 2.326
X-
a
2.32692-75, a=7.309
Z =
0
Scoring below 70, z =
Scoring below 60, z =
70-75
7.309
60-75
7.309
= -0.684
= -2.052
P(score below 70) = 0.2483 (using z = -0.68)
P(score below 60) = 0.0202 (using z = -2.05)
Therefore, 567(0.2483-0.0202) = 129 students scored between 60 and 70.
b) 10 students are good friends. After knowing each other's marks, they came up with a 95%
confidence interval of 65 ± 12 for the population mean. Is their estimation of the population
mean still correct, if they used a 90% confidence level?
The margin of error (+12) is calculated using z-(s = sample standard deviation, n = the number of
Vn
samples, 10). Since remains the same, while the z-score (in university: t-score) shrinks from 1.960 to
1.645
1.645, the margin of error shrinks from 12 to 12-- = 10.07.
1.960
The 90% confidence interval is 65 ± 10.07, where p = 75 is just slightly lower than the upper boundary,
75.07. Therefore, the prediction of the population mean is still correct using a 90% confidence interval.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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