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The article "Career Expert Provides DOs and DON'Ts for Job Seekers on Sodal Networking" (Careerbuilder.com, August 19, 2009 ) included data from a survey of 2667 hiring managers and human resource professionals. The article noted that many employers are using social networks to screen job applicants and that this practice is becoming more common. Of the 2667 people who participared in the survey, 1200 indicated that they use social networking sites (such as Facebook, MySpace, and LinkedIn) to research job applicants. For the purposes of this exercise, assume that the sample is representative of hiring managers and human resource professionals. Construct and interpret a \(95 \%\) confidence interval for the proportion of hiring managers and human resource professionals who use social networking sites to research job applicants.

Step by step solution

01

Identify the sample size and the number of successes

From the problem, the sample size (\(n\)) is 2667, the number of hiring managers and human resource professionals surveyed. The number of successes (\(X\)) is recorded as the number of professionals who use social networking sites to research job applicants, which is 1200.

02

Calculate the sample proportion

The sample proportion (\(\hat{p}\)) is the ratio of the number of successes to the sample size. That is, \(\hat{p} = X/n = 1200 / 2667 ~= 0.4497\)

03

Calculate the standard error

Standard error (SE) for a proportion is given by the square root of \(\hat{p}(1-\hat{p})/n\). Substituting the values, we get SE = \( \sqrt{0.4497 * (1 - 0.4497) / 2667} ~= 0.0093\)

04

Determine the critical value

For a 95% confidence level, the critical value (Z) from the standard normal distribution is approximately 1.96.

05

Construct the confidence interval

The confidence interval is given by \(\hat{p} \pm Z * SE\). Substituting the values we calculated, the confidence interval is \(0.4497 \pm 1.96*0.0093\), which is approximately (0.4315, 0.4679).

06

Interpret the confidence interval

We are 95% confident that the true proportion of hiring managers and human resource professionals who use social networking sites to screen job applicants is between 43.15% and 46.79%.

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Key Concepts

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

Sample Proportion

In the realm of statistics, the sample proportion is a foundational concept when dealing with surveys or samples from a larger population. Essentially, it refers to the ratio of successes to the total sample size. Here, it arises in the context of hiring managers and HR professionals who use social networking sites.
In our exercise, the sample size is 2667, with 1200 individuals indicating they use these sites. To find the sample proportion, you divide the number of successes by the total sample size:

• Formula: \(\hat{p} = \frac{X}{n} \)
• Calculation: \(\hat{p} = \frac{1200}{2667} \approx 0.4497 \)

This means approximately 44.97% of the sampled participants use social networking sites for screening purposes. The sample proportion helps us understand the fraction of the sample that displayed the trait of interest.

Standard Error

The standard error is an indicator of how much the sample proportion is expected to vary from the actual population proportion. It's a measure of the sampling variability and is crucial in constructing confidence intervals.
For a sample proportion, the standard error is calculated using this formula:

• Formula: \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \)
• Calculation: \( SE = \sqrt{\frac{0.4497 \times (1- 0.4497)}{2667}} \approx 0.0093 \)

This implies that the natural fluctuation of the proportion we observe between samples would be around 0.93%. Understanding the standard error helps in assessing how precise our sample proportion is likely to be.

Critical Value

Critical value refers to the cut-off point that corresponds to the desired confidence level in the standard normal distribution. For common confidence levels, such as 95%, the critical value is decided based on the properties of the normal distribution.
For a 95% confidence interval, the critical value (denoted as \( Z \)) is typically 1.96. This value tells us how many standard deviations away from the mean we must go to ensure our range encompasses the middle 95% of our data.

• 95% confidence level: \( Z = 1.96 \)

This number is crucial as it determines the "wiggle room" we allow around our sample estimate to account for sample variability and properly construct our interval.

Normal Distribution

The normal distribution is essential in statistics, especially when exploring concepts like confidence intervals and hypothesis testing. It's a symmetrical, bell-shaped distribution that reflects many natural and social processes. One key feature of the normal distribution is that it allows for the application of the Central Limit Theorem, enabling us to assume that the sampling distribution of the sample proportion is approximately normal, given a large enough sample size, like in our case.

• Key Reason: For large sample sizes, the sampling distribution of the sample mean approaches a normal distribution.
• Application: Normal distribution is used to determine the critical values, such as 1.96 for a 95% confidence interval.

By leveraging properties of the normal distribution, we can make inferences about population parameters even when the population distribution itself is not normal.