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Activity 1: National records have shown that the distribution of time lengths (in years) needed to...


Activity 1: National records have shown that the distribution of time lengths (in years) needed to complete a bachelor's degree is approximately bell-shaped (normal) where the mean is 4.6 yrs and the standard deviation is .3 years.
1C. Would you expect someone to take more than 6 years to complete a bachelor’s degree?
1D. What percent of college students take at most 4.3 years?
1E. What percent of college students take at least 5.2 years?
1F. What percent of students take between 4.3 and 5.2 years?

Solutions

Expert Solution

Given that national records have shown that the distribution of time lengths (in years) needed  to complete a bachelor's degree is approximately bell-shaped (normal) where the mean is 4.6 yrs and the standard deviation is 0.3 years.

Before we go on to solve the problem let us know a bit about normal distribution.

Normal Distribution

A continuous random variable X is said to have a normal distribution if its PDF(Probability Density Function) is given by

its CDF(Cumulative Distribution Function) is given by,

Notation:

Standard Normal Distribution

A continuous random variable X is said to have a standard normal distribution if its PDF(Probability Density Function) is given by

its CDF(Cumulative Distribution Function) is given by,

Exact evaluation of ?(x) is not possible but numerical method can be applied. The values of ?(x) has been tabulated extensively in Biometrika Volume I.

Notation:

Note:

Coming back to our problem,

Here,

X=Distribution of time lengths (in years) needed to complete a bachelor's degree.

1C. Here we need to determine whether we expect someone to take more than 6 years to complete a bachelor’s degree.

[Z~Normal(0,1)]

Hence we do not expect someone to take more than 6 years to complete a bachelor’s degree.

1D. Here we need to determine the percent of college students who take at most 4.3 years to complete a bachelor’s degree.

[Z~Normal(0,1)]

[From Biometrika Tables]

Hence 15.87% college students take at most 4.3 years to complete a bachelor’s degree.

1E. Here we need to determine the percent of college students who take at least 5.2 years to complete a bachelor’s degree.

[Z~Normal(0,1)]

[From Biometrika Tables]

Hence 2.28% college students take at least 5.2 years to complete a bachelor’s degree.

1F. Here we need to determine the percent of college students who take between 4.3 and 5.2 years to complete a bachelor’s degree.

From part 1D.

From part 1E.

Hence 81.85% percent of college students take between 4.3 and 5.2 years to complete a bachelor’s degree.


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