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We are interested in estimating the proportion of graduates at a mid-sized university who found a job within one year of completing their undergraduate degree. Suppose we conduct a survey and find out that 348 of the 400 randomly sampled graduates found jobs. The graduating class under consideration included over 4500 students. (a) Describe the population parameter of interest. What is the value of the point estimate of this parameter? (b) Check if the conditions for constructing a confidence interval based on these data are met. (c) Calculate a \(95 \%\) confidence interval for the proportion of graduates who found a job within one year of completing their undergraduate degree at this university, and interpret it in the context of the data. (d) What does "95\% confidence" mean? (e) Now calculate a \(99 \%\) confidence interval for the same parameter and interpret it in the context of the data. (f) Compare the widths of the \(95 \%\) and \(99 \%\) confidence intervals. Which one is wider? Explain.

Step by step solution

01

Identify the population parameter

The population parameter of interest is the proportion of graduates from the university who found a job within one year of completing their undergraduate degree. The point estimate of this parameter, based on the sample, is calculated as the number of graduates who found jobs divided by the total number of surveyed graduates: \( \hat{p} = \frac{348}{400} = 0.87 \).

02

Check conditions for confidence interval

To construct a confidence interval, the sample should be random, the population should be at least 10 times the sample size, and both \(np\) and \(n(1-p)\) should be at least 10. The sample is random, the population (4500) is over 10 times the sample size (400), and \(np = 348\) and \(n(1-p) = 52\) which are both greater than 10. Thus, conditions are satisfied.

03

Calculate 95% Confidence Interval

A 95% confidence interval for a proportion is calculated by \( \hat{p} \pm z \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } \) where \( z = 1.96 \) for 95% confidence. Here, \( \hat{p} = 0.87 \) and \( n = 400 \). The interval is \( 0.87 \pm 1.96 \times \sqrt{ \frac{0.87 \times 0.13}{400} } \). Calculating gives an interval from approximately 0.837 to 0.903.

04

Interpret the 95% Confidence Interval

We are 95% confident that the true proportion of all graduates from this university who found a job within one year is between 83.7% and 90.3%.

05

Explain 95% confidence

A 95% confidence level means that if we took many samples and built a confidence interval from each of them, we would expect about 95% of those intervals to contain the true population proportion.

06

Calculate 99% Confidence Interval

For a 99% confidence interval, we use \( z = 2.576 \). The interval is \( 0.87 \pm 2.576 \times \sqrt{ \frac{0.87 \times 0.13}{400} } \). Calculating gives an interval from approximately 0.826 to 0.914.

07

Interpret the 99% Confidence Interval

We are 99% confident that the true proportion of all graduates from this university who found a job within one year is between 82.6% and 91.4%.

08

Compare 95% and 99% Confidence Intervals

The 99% confidence interval is wider than the 95% confidence interval. This is because greater confidence requires accounting for more uncertainty, thus a wider interval.

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Key Concepts

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

Point Estimate

In statistics, the point estimate serves as a single value used to approximate an unknown population parameter. For the scenario involving university graduates, we're specifically looking at the proportion of graduates who secured a job within a year.

This is represented as \(\hat{p}\), the sample proportion. In the given problem, we had 348 graduates out of 400 who found jobs. Thus, the point estimate for the proportion is:\[\hat{p} = \frac{348}{400} = 0.87\]

This means our best guess for the actual proportion of all graduates finding a job is 0.87, or 87%.

Population Parameter

The population parameter is a value that describes a characteristic of a population; in our case, it refers to the proportion of all university graduates who find a job within a year of graduation. While we aim to estimate this parameter, we usually don't know its true value.

Due to practical constraints, like time and resources, it's uncommon to measure an entire population directly. Hence, we rely on data from a sample (a small number of the population) to make an educated guess about the parameter, using a point estimate. Importantly, the population parameter is what we seek to estimate accurately through our sample data.

Sample Size

The sample size is crucial when analyzing data, as it affects the precision of our estimates. In this exercise, the sample size refers to the 400 randomly selected graduates from the university.

A larger sample size usually offers more reliable estimates, providing a clearer picture of the whole population. Also, a larger sample better satisfies the conditions necessary for constructing a confidence interval. The essential conditions include:

• The sample must be random.
• Population should be at least 10 times the sample size.
• Both \( np \) and \( n(1-p) \) should be at least 10 to satisfy the Normal approximation condition.

With our scenario, we fulfill all these conditions, allowing us to proceed with accurate confidence interval calculations.

Proportion

A proportion represents a part or fraction of a whole, often expressed as a percentage. It's a useful statistic for summarizing the result of categorical data.

In this exercise, the proportion of interest is the fraction of university graduates who have secured a job within one year. Calculating this provides a comprehensive view of the employability of graduates from this specific institution.

This proportion can lead to an insightful analysis, such as comparing different cohorts or assessing changes over time. It's fundamental to many statistical investigations as it highlights crucial dimensions of the dataset sequences, like our 87% estimate for job-acquiring graduates from this exercise.