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Chapter 10: Problem 7

According to a 2017 report, 53\% of college graduates in California had student loans. Suppose a random sample of 120 college graduates in California shows that 72 had college loans. (Source: Lendedu.com) a. What is the observed frequency of college graduates in the sample who had student loans? b. What is the observed proportion of college graduates in the sample who had student loans? c. What is the expected number of college graduates in the sample to have student loans if \(53 \%\) is the correct rate? Do not round off.

Short Answer

Expert verified
a. The observed frequency of college graduates in the sample who had student loans is 72.\nb. The observed proportion is \(0.6\) (or \(60\%\) if given as a percentage).\nc. The expected number of graduates to have student loans if \(53\%\) is the correct rate is \(63.6\).

Step by step solution

01

Calculation of Observed Frequency

The observed frequency is the raw count of events we observe in our sample. In this case, it refers to the number of college graduates who had student loans. According to the problem, this is given as 72.
02

Calculation of Observed Proportion

The observed proportion is the frequency of an event divided by the total number of trials or observations. In this exercise, the observed proportion of college graduates who had student loans can be calculated by dividing the observed frequency (number of students with loans) by the total number of students (sample size). In other words, \(Observed Proportion = \frac{Observed Frequency}{Sample Size} = \frac{72}{120}\).
03

Calculation of Expected Number

The expected number refers to the predicted count of events based on a known rate. Here, we know that 53% of college graduates have student loans. To find the expected number of students in our sample who would have loans if this rate is accurate, we simply multiply the sample size by the rate, which can be interpreted as a probability: \(Expected Number = Sample Size * Rate = 120 * 0.53\).

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Key Concepts

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

Observed Frequency
When diving into statistics, one of the first concepts you'll encounter is the observed frequency. Simply put, observed frequency is the count of occurrences you note in your sample data. It's like tallying the marks you make each time a specific event occurs. In our given exercise about college graduates in California, the observed frequency refers to the number of graduates who have student loans. We've been provided with the information that out of 120 graduates, 72 had loans. Think of it as a simple headcount. This number, 72, represents our observed frequency for students in debt. Observed frequency helps us gain a concrete sense of what happens in a sample, acting as the foundation for deeper statistical analysis.
Observed Proportion
The observed proportion takes us a step further by converting our raw counts into ratios. Why do we do this? Well, proportions allow us to understand observed data in relation to the whole, which is incredibly useful when comparing different groups or sets of data. In this exercise, we calculate the observed proportion of college graduates with student loans by dividing the observed frequency (72 students with loans) by the total number of students in the sample (120). This gives us the formula:
  • \(Observed \ Proportion = \frac{72}{120}\)
When calculated, this ratio translates to 0.6 or 60%, which tells us that 60% of the sampled students have loans. Proportions help to normalize data, making it more straightforward to compare different-sized groups or over different periods.
Expected Number
Estimating the expected number is a vital step in statistical analysis. It's like guessing how many beans you're likely to find in a jar based on a sample taste. The expected number gives us an idea of what to anticipate under a known condition. For our college loans scenario, where 53% of graduates typically have loans, the expected number helps us compare actual data against this backdrop.
  • To calculate it, multiply the total sample by the given rate or probability of the event.
  • Here's the formula: \(Expected \ Number = 120 \times 0.53\)
This results in an expected number of 63.6 graduates having loans. While you can't have a fraction of a graduate, this tells us statistically about how many, on average, are expected if the 53% rate holds true for our sample size.

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Most popular questions from this chapter

Rats had a choice of freeing another rat or eating chocolate by themselves. Most of the rats freed the other rat and then shared the chocolate with it. The table shows the data concerning the gender of the rat in control. $$\begin{array}{lcc} & \text { Male } & \text { Female } \\\\\hline \text { Freed Rat } & 17 & 6 \\\\\text { Did not } & 7 & 0 \\ \hline\end{array}$$ a. Can a chi-square test for homogeneity or independence be performed with this data set? Why or why not? b. Determine whether the sex of a rat influences whether or not it frees another rat using a significance level of \(0.05\).

In the study described in \(10.35\) researchers also asked survey respondents if they had heard of the HPV vaccine. Data are shown in the table. Test the hypothesis that knowledge of the vaccine and race are associated. Use a \(0.05\) significance level. $$\begin{array}{lcc}\hline \text { Heard of HPV vaccine } & \text { AAPI } & \text { White } \\ \hline \text { Yes } & 248 & 1737 \\\\\text { No } & 103 & 193 \\\\\hline\end{array}$$

The table shows the percentage of all men and women in the United States aged 18 to 44 who meet aerobic fitness guidelines. Give two reasons why a chi- square test is not appropriate for this data. $$\begin{array}{|c|c|c|} \hline & {\text { Percentage Meeting Fitness Guidelines }} \\\\\hline \text { Year } & \text { Men } & \text { Women } \\ \hline 2005 & 50.0 & 43.1 \\ \hline 2010 & 59.0 & 48.5 \\ \hline 2014 & 60.8 & 52.7 \\ \hline\end{array}$$

Suppose you randomly assign some parolees to check in once a week with their parole officers and others to check in once a month, and observe whether they are arrested within 6 months of starting parole.

The Perry Preschool Project data presented in exercise \(10.39\) (Schweinhart et al. 2005 ) can be divided to see whether the preschool attendance effect is different for males and females. The table shows a summary of the data for females, and the figure shows Minitab output that you may use. $$\begin{array}{|lcc|}\hline & \text { Preschool } & \text { No Preschool } \\\\\hline \text { HS Grad } & 21 & 8 \\\\\hline \text { HS Grad No } & 4 & 17 \\\ \hline\end{array}$$ a. Find the graduation rate for those females who went to preschool, and compare it with the graduation rate for females who did not go to preschool. b. Test the hypothesis that preschool and graduation rate are associated, using a significance level of \(0.05\).
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