Textbook Question
Using and Interpreting Concepts
Finding the Range of a Data Set In Exercises 9 and 10, find the range of the data set represented by the graph.
3. Describing Data Numerically
Standard Deviation
1:38 minutesProblem 2.4.20
Textbook Question
Salary Offers You are applying for jobs at two companies. Company C offers starting salaries with μ = $59,000 and σ = $1500. Company D offers starting salaries with μ = $59,000 and σ = $1000. From which company are you more likely to get an offer of $62,000 or more? Explain your reasoning.
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Step 1: Recognize that the problem involves comparing probabilities of getting a salary offer of $62,000 or more from two companies. This requires calculating the z-scores for $62,000 for both companies, as the z-score standardizes the value relative to the mean and standard deviation of each distribution.
Step 2: Use the z-score formula: z = (X - μ) / σ, where X is the value of interest ($62,000), μ is the mean salary, and σ is the standard deviation. For Company C, substitute μ = 59,000 and σ = 1,500 into the formula.
Step 3: Similarly, calculate the z-score for Company D using the same formula, but substitute μ = 59,000 and σ = 1,000.
Step 4: Once the z-scores are calculated for both companies, use a standard normal distribution table (or a statistical software) to find the probability corresponding to each z-score. Since the problem asks for $62,000 or more, calculate the area to the right of the z-score (1 - cumulative probability).
Step 5: Compare the probabilities for both companies. The company with the smaller probability of $62,000 or more is less likely to offer that salary, while the company with the larger probability is more likely. Explain your reasoning based on the spread (standard deviation) of the distributions.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In this context, both companies' salary offers can be modeled as normal distributions, characterized by their mean (μ) and standard deviation (σ). Understanding this concept is crucial for determining the likelihood of receiving a salary offer above a certain threshold.
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Z-Score
A Z-score indicates how many standard deviations an element is from the mean. It is calculated using the formula Z = (X - μ) / σ, where X is the value of interest. By calculating the Z-scores for the $62,000 salary offer for both companies, we can compare how likely it is to receive such an offer relative to each company's salary distribution.
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Probability and Area Under the Curve
In the context of normal distributions, the probability of a certain outcome can be determined by calculating the area under the curve of the distribution. This area represents the likelihood of receiving a salary offer of $62,000 or more. By using Z-scores to find the corresponding probabilities from standard normal distribution tables, we can assess which company is more likely to provide a higher salary offer.
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