Q. 6.109 - Elementary Statistics

Question

College-Math Success. Researchers S. Lesik and M. Mitchell explore the difficulty of predicting success in college-level mathematics in the article "The Investigation of Multiple Paths to Success in College-Level Mathematics" (fraternal of Applied Reacuwh in Hreher Eiturarion, Vol. 5. Issue 1. pP, 48-57). One of the variables explored as an indicator of success was the length of time since a college freshman has taken a mathematics course. The article reports that the mean length of time is 0.18 years with a standard deviation of 0.624 years. For college freshmen, let x represent the time, in years, since taking a math course.

A . What percentage of times are at least 0 years?

b. Assuming that x is approximately normally distributed, tose normal curve areas to determine the approximate percentage of times that are at least 0 years.

c. Based on your results from parts (a) and (b), do you think that the length of time since taking a math course for college freshmen is approximately a normally distributed variable? Explain your answer.

Step-by-Step Solution

Answer

a) $100\%$

b) $61.41\%$

c) No

Part (a) Step 1: Given Information

To find The percentage of time lengths that are at least 0 years.

Part (a) Step 2: Explanation

Given the following information:

The time it takes a college freshman to complete a mathematics course $\left(x\right)$ has a mean of $\mu =0.18$ years and a standard deviation of $\sigma =0.624$ years.

It is not possible to have a negative time length based on the information provided. As a result, the time span is always greater than or equal to $0$ years. As a result, the percentage of time lengths of at least $0$ years is $100 \%$

Part (b) Step 1: Given Information

If x is roughly followed by a normal distribution, find the percentage of time lengths that are at least 0 years.

Part (b) Step 2: Explanation

Determine the z- scores :

$$\begin{gathered} z=\frac{0-0.18}{0.624} \\ \approx -0.29 \end{gathered}$$

Determine the corresponding probability using the normal probability table in Appendix A:

$P(x>0)=P(z>-0.29)$

              $=1-0.3859$

              $=0.6141$

              $=61.41\%$

Part (c) Step 1: Given Information

To explain Whether  x is approximately follows a normal distribution from the results of part (a) and (b). }

Part (c) Step 2: Explanation

The percentages in parts (a) and (b) above should be equal to $100\%,$ but the results of part (c) differ significantly from (b). As a result, the conclusion that $x$ roughly follows a normal distribution cannot be reached.