
To find:
To write is there sufficient evidence to conclude at

Answer to Problem 10P
Solution:
There is no sufficient evidence to conclude that the makeup of your institution has significantly changed between Year 1 and Year 2.
Explanation of Solution
Consider two years that are not consecutive from which to collect data.
That is choosing the current academic year for Year 2 and the academic year four years earlier for Year 1.
Suppose the 2015 for year 1 and 2019 for year 2.
For year 1:
The number of students who were enrolled at your institution for each classification during year 1 is given below,
2015 | |
Number of Freshmen Enrolled | 15 |
Number of Sophomores Enrolled | 18 |
Number of Juniors Enrolled | 21 |
Number of Seniors Enrolled | 73 |
Total Number of Students Enrolled | 127 |
Let
Mathematically, we can write the null and alternative hypothesis for four different students as follows,
Let‘s calculate the
And,
For year 2:
The number of students who were enrolled at your institution for each classification during year 2 is given below,
2015 | |
Number of Freshmen Enrolled | 24 |
Number of Sophomores Enrolled | 20 |
Number of Juniors Enrolled | 19 |
Number of Seniors Enrolled | 20 |
Let
Mathematically, we can write the null and alternative hypothesis for four different students as follows,
2019 | |
Percentage of Freshmen Enrolled | 28.92% |
Percentage of Sophomores Enrolled | 24.10% |
Percentage of Juniors Enrolled | 22.89% |
Percentage of Seniors Enrolled | 24.10% |
Let‘s calculate the expected value for year 2. Since we are assuming that the proportion percentages stated for the student in year 2 are not incorrect is calculated as follows,
And,
Formula for calculating test statistic for a Chi-Square test for goodness of fit:
The test statistic for a chi-square test for goodness of fit is given below,
Where Oi, is the observed frequency for the ith possible outcome and Ei is the expected frequency for the ith possible outcome and n is the sample size.
From the above table represent the value of n for year 1 is 127 and for year 2 is 83.
The following table represents the test statistic,
Year 1 | |||||
Category | Observed Values | Expected values | |||
Freshman | 15 | 14.9987 | 0.0013 | ||
Sophomore | 18 | 17.9959 | 0.0041 | ||
Junior | 21 | 21.0058 | -0.0058 | ||
Senior | 73 | 72.9996 | 0.0004 | ||
Year 2 | |||||
Freshman | 24 | 24.0036 | -0.0036 | ||
Sophomore | 20 | 20.003 | -0.003 | ||
Junior | 19 | 18.9987 | 0.0013 | ||
Senior | 20 | 20.003 | -0.003 | ||
The calculated test statistic for year 1 and for year 2 is
Degrees of freedom in a Chi-square test for goodness of fit:
In a chi-square test for goodness of fit the number of degrees of freedom for the chi-square distribution of the test statistic is given by,
Where k is the number of possible outcomes for each trial.
Rejection Region for Chi-Square Test for Association:
Reject the null hypothesis,
From the given information k = 4.
Substitute the above values in the formula of degrees of freedom to get the following,
Conclusion:
Use the level of significance of
Use the “Area to the right of the critical value
Year 1:
The test statistic value is
By using the above condition we fail to reject the null hypothesis.
Since
Year 2:
The test statistic value is
By using the above condition we fail to reject the null hypothesis.
Since
The null hypothesis will be accepted for both year 1 and year 2.
Hence the proportion percentage for the students does not vary for both year 1 and year 2.
So, there is no sufficient evidence to conclude that the makeup of your institution has significantly changed between Year 1 and Year 2. Since the proportion percentage will not vary between two years. That is, percentage of students in every classification in year 1will be equal to percentage of students in every classification in year 2.
Final statement:
There is no sufficient evidence to conclude that the makeup of your institution has significantly changed between Year 1 and Year 2.
Want to see more full solutions like this?
Chapter 10 Solutions
Beginning Statistics, 2nd Edition
- Theorem 5.1 (Jensen's inequality) state without proof the Jensen's Ineg. Let X be a random variable, g a convex function, and suppose that X and g(X) are integrable. Then g(EX) < Eg(X).arrow_forwardCan social media mistakes hurt your chances of finding a job? According to a survey of 1,000 hiring managers across many different industries, 76% claim that they use social media sites to research prospective candidates for any job. Calculate the probabilities of the following events. (Round your answers to three decimal places.) answer parts a-c. a) Out of 30 job listings, at least 19 will conduct social media screening. b) Out of 30 job listings, fewer than 17 will conduct social media screening. c) Out of 30 job listings, exactly between 19 and 22 (including 19 and 22) will conduct social media screening. show all steps for probabilities please. answer parts a-c.arrow_forwardQuestion: we know that for rt. (x+ys s ا. 13. rs. and my so using this, show that it vye and EIXI, EIYO This : E (IX + Y) ≤2" (EIX (" + Ely!")arrow_forward
- Theorem 2.4 (The Hölder inequality) Let p+q=1. If E|X|P < ∞ and E|Y| < ∞, then . |EXY ≤ E|XY|||X|| ||||qarrow_forwardTheorem 7.6 (Etemadi's inequality) Let X1, X2, X, be independent random variables. Then, for all x > 0, P(max |S|>3x) ≤3 max P(S| > x). Isk≤narrow_forwardTheorem 7.2 Suppose that E X = 0 for all k, that Var X = 0} x) ≤ 2P(S>x 1≤k≤n S√2), -S√2). P(max Sk>x) ≤ 2P(|S|>x- 1arrow_forwardThree players (one divider and two choosers) are going to divide a cake fairly using the lone divider method. The divider cuts the cake into three slices (s1, s2, and s3).If the chooser's declarations are Chooser 1: {s3} and Chooser 2: {s3}, which of the following is a fair division of the cake?arrow_forwardTheorem 1.4 (Chebyshev's inequality) (i) Suppose that Var X x)≤- x > 0. 2 (ii) If X1, X2,..., X, are independent with mean 0 and finite variances, then Στη Var Xe P(|Sn| > x)≤ x > 0. (iii) If, in addition, X1, X2, Xn are identically distributed, then nVar Xi P(|Sn> x) ≤ x > 0. x²arrow_forwardTheorem 2.5 (The Lyapounov inequality) For 0arrow_forwardTheorem 1.6 (The Kolmogorov inequality) Let X1, X2, Xn be independent random variables with mean 0 and suppose that Var Xk 0, P(max Sk>x) ≤ Isk≤n Σ-Var X In particular, if X1, X2,..., X, are identically distributed, then P(max Sx) ≤ Isk≤n nVar X₁ x2arrow_forwardTheorem 3.1 (The Cauchy-Schwarz inequality) Suppose that X and Y have finite variances. Then |EXYarrow_forwardAbout 25% of people in America use a certain social media website. In a group with 20 people (assume that it is a random sample of people in America), what are the following probabilities? (Round your answers to three decimal places.) a) At least one of them uses the website. b) More than two of them use the website. c) None of them use the website. d) At least 18 of them do not use the website. please show all steps and work for probabilities. answer parts a-d.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
- MATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th...StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C...StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage Learning
- Elementary Statistics: Picturing the World (7th E...StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman
MATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th...StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C...StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E...StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman