/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 A mathematics placement test is ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 6

A mathematics placement test is given to all entering freshmen at a small college. A student who receives a grade below 35 is denied admission to the regular mathematics course and placed in a remedial class. The placement test scores and the final grades for 20 students who took the regular course were recorded as follows: $$ \begin{array}{cc} \text { Placement Test } & \text { Course Grade } \\ \hline 50 & 53 \\ 35 & 41 \\ 35 & 61 \\ 40 & 56 \\ 55 & 68 \\ 65 & 36 \\ 35 & 11 \\ 60 & 70 \\ 90 & 79 \\ 35 & 59 \\ 90 & 54 \\ 80 & 91 \\ 60 & 48 \\ 60 & 71 \\ 60 & 71 \\ 40 & 47 \\ 55 & 53 \\ 50 & 68 \\ 65 & 57 \\ 50 & 79 \end{array} $$ (a) Plot a scatter diagram. (b) Find the equation of the regression line to predict course grades from placement test scores. (c) Graph the line on the scatter diagram. (d) If 60 is the minimum passing grade, below which placement test score should students in the future be denied admission to this course?

Short Answer

Expert verified
After implementing these steps, you should have a visual representation of the correlation between placement test scores and final course grades in a scatter plot. The equation of the regression line will aid in the prediction of final grades based on placement test scores. Using this, the test score below which students should be denied admission, can be found by setting the course grade to 60 in the regression equation and solving for the placement test score.

Step by step solution

01

Plotting a scatter diagram

List all the pair of scores (Placement Test, Course Grade) as coordinates (x, y) in a scatter plot.
02

Calculating the regression line

First, calculate the mean of the Placement Test Scores (x) and Course Grade (y). Then compute the difference between each score and its mean, multiplying the differences for each pair of x and y. Sum all these products while also summing the square of the difference for x. The slope (b) of the line can be found by dividing the two sums. Find the intercept (a) by subtracting the product of the mean of x and b from the mean of y.
03

Plotting the regression line

Use the equation y = ax + b, substitute x with the Placement Test Scores to get the predicted Course Grade. Plot this line on the scatter diagram.
04

Predicting the threshold value

Set y in the equation to 60 (minimum passing grade) and solve for x. This will give the Placement Test Score below which students should be denied admission.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatter Plot
Scatter plots are visual tools used to display and assess the relationship between two continuous variables. In our example, each student's placement test score is paired with their course grade and represented as a point on the graph. The x-axis displays the placement test scores, while the y-axis portrays the course grades. Each point's position reflects the student's performance in both areas.

Scatter plots help in understanding how one variable affects another at a glance. By plotting the placement test scores against course grades, we can investigate whether a higher test score results in a higher course grade. This helps in identifying patterns and relationships.
  • Identify clusters of points that suggest a relationship.
  • Observe any outliers or unusual data points that deviate from the cluster.
  • Notice the overall trend direction if any is present.
Statistical Correlation
The statistical correlation quantifies the strength and direction of the relationship between two variables. It ranges from -1 to 1, where values closer to 1 or -1 indicate a strong relationship, and values near 0 suggest a weak or no relationship. In our case, we are interested in how closely linked the placement tests are to the course grades.

Correlation is typically measured using Pearson's correlation coefficient, represented by the symbol \( r \). A positive \( r \) indicates that as one variable increases, the other does too, which may be seen if better placement scores lead to better final grades. Conversely, a negative \( r \) denotes an inverse relationship.
  • Values near +1: Strong positive relationship (e.g., high test scores often mean high grades).
  • Values near -1: Strong negative relationship (rare in our context).
  • Values around 0: Little to no observable relationship.
Recognizing these patterns helps in understanding the predictive power when we build models based on regression analysis.
Predictive Modeling
Predictive modeling uses statistical techniques to develop a mathematical model that can be used to predict future outcomes. In the context of our problem, a regression line was developed to predict course grades based on placement test scores.

Regression analysis helps in drawing the best-fit line through the data points plotted on a scatter diagram. This line is represented by the linear equation \( y = ax + b \), where \( a \) is the slope and \( b \) is the y-intercept. It indicates how much the course grade is expected to change with each additional point on the placement test score.
  • Helps in making informed decisions like setting a threshold for test scores.
  • Enhances understanding of the relationship between variables.
  • Provides a quantified basis for predictions which can be validated with past data.
Hence, predictive modeling not only sheds light on current relationships but also equips us to make reliable predictions for the future based on data trends.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For a particular variety of plant, researchers wanted to develop a formula for predicting the quantity of seeds (grams) as a function of the density of plants. They conducted a study with four levels of the factor \(X\), the number of plants per plot. Four replications were used for each level of \(X .\) The data are shown as follows: $$ \begin{array}{ccccc} \text { Plants per Plot } && {\text { Quantity of Seeds, } y} \\ \ {X} && \ {\text { (grams) }} \\ \hline 10 & &12.6 & 11.0 & \mathbf{1 2 . 1} & 10.9 \\ 20 && 15.3 & 16.1 & 14.9 & 15.6 \\ 30 && 17.9 & 18.3 & 18.6 & 17.8 \\ 40 & &19.2 & 19.6 & 18.9 & 20.0 \end{array} $$ Is a simple linear regression model adequate for analyzing this data set?

(a) Find the least squares estimate for the parameter \(\beta\) in the linear equation \(\mu_{Y \mid x}=3 x\). (b) Estimate the regression line passing through the origin for the following data: $$ \begin{array}{c|cccccc} x & 0.5 & 1.5 & 3.2 & 4.2 & 5.1 & 6.5 \\ \hline y & 1.3 & 3.4 & 6.7 & 8.0 & 10.0 & 13.2 \end{array} $$

A study of the amount of rainfall and the quantity of air pollution removed produced the following data: $$ \begin{array}{cc} \text { Daily Rainfall, } & \text { Particulate Removed, } \\ x(0.01 \mathrm{~cm}) & y\left(\mu \mathrm{g} / \mathrm{m}^{3}\right) \\ \hline 4.3 & 126 \\ 4.5 & 121 \\ 5.9 & 116 \\ 5.6 & 118 \\ 6.1 & 114 \\ 5.2 & 118 \\ 3.8 & 132 \\ 2.5 & 141 \\ 7.5 & 108 \end{array} $$ (a) Find the equation of the regression line to predict the particulate removed from the amount of daily rainfall. (b) Estimate the amount of particulate rernoved when the daily rainfall is \(x=4.8\) units.

Heat, treating is often used to carburize metal parts such as gears. The thickness of the carburized layer is considered an important feature of the gear, and it contributes to the overall reliability of the part. Because of the critical nature of this feature, a lab test is performed on each furnace load. The test is a destructive one, where an actual part is cross sectioned and soaked in a chemical for a period of time. This test involves running a carbon analysis on the surface of both the gear pitch (top of the gear tooth) and the gear root (between the gear teeth). The data below are the results of the pitch carbon-analysis test catch for 19 parts. $$ \begin{array}{cccc} \text { Soak Time } & \text { Pitch } & \text { Soak Time } & \text { Pitch } \\\ \hline 0.58 & 0.013 & 1.17 & 0.021 \\ 0.66 & 0.016 & 1.17 & 0.019 \\ 0.66 & 0.015 & 1.17 & 0.021 \\ 0.66 & 0.016 & 1.20 & 0.025 \\ 0.66 & 0.015 & 2.00 & 0.025 \\ 0.66 & 0.016 & 2.00 & 0.026 \\ 1.00 & 0.014 & 2.20 & 0.024 \\ 1.17 & 0.021 & 2.20 & 0.025 \\ 1.17 & 0.018 & 2.20 & 0.024 \\ 1.17 & 0.019 & & \end{array} $$ (a) Fit a simple linear regression relating the pitch carbon analysis \(y\) against soak time. Test \(H_{0}: \beta_{1}=0\). (b) If the hypothesis in part (a) is rejected, determine if the linear model is adequate.

Physical fitness testing is an important aspect of athletic training. A common measure of the magnitude of cardiovascular fitness is the maximum volume of oxygen uptake during a strenuous exercise. A study was conducted on 24 middleaged men to study the influence of the time it takes to complete a two mile run. The oxygen uptake measure was accomplished with standard laboratory methods as the subjects performed on a treadmill. The work was published in "Maximal Oxygen Intake Prediction in Young and Middle Aged Males," Journal of Sports Medicine 9 , \(1969,17-22 .\) The data are as presented here. $$ \begin{array}{ccc} & y \text { , Maximum } & \text { x, Time } \\ \text { Subject } & \text { Volume of } \mathbf{O}_{2} & \text { in } \text { Seconds } \\ \hline 1 & 42.33 & 918 \\ 2 & 53.10 & 805 \\ 3 & 42.08 & 892 \\ 4 & 50.06 & 962 \\ 5 & 42.45 & 968 \end{array} $$ (Figure) $$ \begin{array}{ccc} & y, \text { Maximum } & x, \text { Time } \\ \text { Subject } & \text { Volume of } \mathrm{O}_{2} & \text { in } \text { Seconds } \\ \hline \mathbf{6} & 42.46 & 907 \\ 7 & 47.82 & 770 \\ \mathbf{8} & 49.92 & 743 \\ \mathbf{9} & 36.23 & 1045 \\ 10 & 49.66 & 810 \\ 11 & 41.49 & 927 \\ 12 & 46.17 & 813 \\ 13 & 46.18 & 858 \\ 14 & 43.21 & 860 \\ 15 & 51.81 & 760 \end{array} $$ $$ \begin{array}{ccc} & y \text { , Maximum } & x \text { , Time } \\ \text { Subject } & \text { Volume of } \mathrm{O}_{2} & \text { in } \text { Seconds } \\ \hline 16 & 53.28 & 747 \\ 17 & 53.29 & 743 \\ 18 & 47.18 & 803 \\ 19 & 56.91 & 683 \\ 20 & 47.80 & 844 \\ 21 & 48.65 & 755 \\ 22 & 53.67 & 700 \\ 23 & 60.62 & 748 \\ 24 & 56.73 & 775 \end{array} $$ (Figure) (a) Estimate the parameters in a simple linear regression model. (b) Does the time it takes to run two miles have a significant influence on maximum oxygen uptake? Use \(\begin{array}{ll}H_{0}: & 0=0 \\ H_{1}: & \beta \neq 0\end{array}\) (c) Plot the residuals on a graph against \(x\) and comment on the appropriateness of the simple linear model.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.