Paper For Above instruction
Problem 1: Correlation between Absences and Final Scores
An educator seeks to determine how a student's number of absences affects their final score. Given a dataset of paired observations, assume a linear relationship exists between these variables. Calculate the Pearson correlation coefficient (r) to quantify the strength and direction of this linear relationship.
Suppose the data points are as follows:
Number of Absences: 0, 2, 4, 6, 8
Final Scores: 88, 76, 70, 65, 60
To calculate the Pearson correlation coefficient, use the formula: r = ∑((x i - x■)(y i
Calculations:
Compute the means x■ and ■
Calculate deviations and their products.
Sum the products and the squared deviations.
Apply the formula to find r.
The resulting correlation coefficient indicates a strong negative linear relationship between absences and final scores, illustrating that as absences increase, scores tend to decrease.
Problem 2: Probability of Business Job Eliminations
Past data indicates that 13% of businesses have eliminated jobs over the past year. If five businesses are selected at random, determine the probability that at least three have eliminated jobs during this period.
Using the binomial probability formula:
P(X ≥ 3) = P(X=3) + P(X=4) + P(X=5)
Where:
n = 5 (number of trials)
p = 0.13 (probability of success, i.e., business eliminated jobs)
Calculate each term using:
P(X=k) = C(n,k) * p k * (1 - p)
Sum these probabilities to find the total probability that at least three businesses eliminated jobs.
The resulting probability reflects the likelihood of observing three or more businesses with job eliminations out of five randomly selected companies.
Problem 3: Home Values and Probability
A study of 800 homes in an area found that the average home value was \$82,000 with a standard deviation of \$5,000. If 50 homes are for sale, what is the probability that their sample mean value exceeds \$83,500?
Assuming normality, the sampling distribution of the sample mean has:
Mean = \$82,000
Standard error = σ / √n = 5000 / √50 ≈ 707.11
Calculate the z-score:
z = (X■ - µ) / (σ / √n) = (83500 - 82000) / 707.11 ≈ 2.12
Using standard normal distribution tables or calculator, find P(Z > 2.12) ≈ 0.017
Thus, there is approximately a 1.7% chance that the mean value of the 50 homes exceeds \$83,500.
Problem 4: Chi-Square Goodness-of-Fit Test
The college's dean hypothesizes the distribution of students across different majors aligns with specified percentages:
Business (BU): 40%
Computer Science (CS): 25%
Science (SC): 15%
Social Science (SS): 10%
Liberal Arts (LA): 5%
General Studies (GS): 5%
Actual enrollment last semester was:
CS: 130
SC: 60
SS: 50
LA: 20
GS: 20
Perform a Chi-Square goodness-of-fit test at α=0.10 to determine if the observed distribution significantly differs from the hypothesized percentages. Steps include:
Calculating expected frequencies based on total enrollment.
Computing chi-square statistic: χ² = ∑ (O - E)² / E.
Comparing with critical value from χ² distribution table with degrees of freedom = number of categories1 = 5.
Conclusion: Based on the calculated χ² and critical value, determine whether to accept or reject the null hypothesis that the distribution matches.
Problem 5: T-Test for Mean Scores in Different Majors
A statistician claims that psychology students score higher on a test than mathematics students. Sample data:
Psychology: n=84, mean=61.2, SD=7.9
Mathematics: n=34, mean=59.4, SD=7.9
Test the claim at α=0.01 by performing a two-sample t-test with unequal variances (Welch's t-test). Steps include:
State hypotheses: H■: µ psych
math
, H■: µ
psych
> µ
math
. Calculate the t-statistic:
t = (mean
psych - mean
math
) / √(s■²/n■ + s■²/n■)
Calculate degrees of freedom using Welch-Satterthwaite equation.
Compare calculated t to critical t-value for df at α=0.01.
If t > t
critical
, reject H■, supporting the claim that psychology scores are higher.
Problem 6: Variance and Standard Deviation of a Distribution
The probability distribution of the number of people on hold when calling a radio station is:
Calculate the expected value, variance, and standard deviation of X. Then, assess whether increasing the number of phone lines would be beneficial based on the variability in phone calls.
Calculations:
Expected value: E[X] = ∑ x * P(X=x)
Variance: Var(X) = ∑ (x - E[X])² * P(X=x)
Standard deviation: σ = √Var(X)
Conclusion: If the variance and standard deviation suggest high variability, consider whether more lines would reduce call dropping or busy signals.
References
Agresti, A. (2018). Statistical Methods for the Social Sciences. Pearson.
Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W. H. Freeman.
Ross, S. (2014). Introduction to Probability Models. Academic Press.
Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
Wackerly, D. D., Mendenhall, W., & Scheaffer, R. L. (2008). Mathematical Statistics with Applications.
Cengage Learning.
Triola, M. F. (2018). Elementary Statistics. Pearson. Schervish, M. J. (2012). Theory of Statistics. Springer.
Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis. Brooks/Cole.
Vaughan, R. (2020). Data Analysis in Social Science. Sage Publications.
