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Paired or not? In each of the following scenarios, determine if the data are paired. (a) We would like to know if Intel's stock and Southwest Airlines' stock have similar rates of return. To find out, we take a random sample of 50 days for Intel's stock and another random sample of 50 days for Southwest's stock. (b) We randomly sample 50 items from Target stores and note the price for each. Then we visit Walmart and collect the price for each of those same 50 items. (c) A school board would like to determine whether there is a difference in average SAT scores for students at one high school versus another high school in the district. To check, they take a simple random sample of 100 students from each high school.

Step by step solution

01

Analyze Scenario (a)

Determine if the stock data for Intel and Southwest Airlines are paired by examining how the samples are taken. A separate random sample of 50 days is taken from Intel and another separate sample from Southwest Airlines without any direct connection between the samples. Thus, the samples are not paired.

02

Analyze Scenario (b)

Determine the relationship between data from Target and Walmart. Here, identical items are priced at both stores, establishing a direct connection for each pricing pair across the two stores. Therefore, the data are paired.

03

Analyze Scenario (c)

Examine the sampling method for SAT scores from two high schools. The samples are taken independently from two schools with no direct matching of individuals or exam instances, hence the data are not paired.

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Key Concepts

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

Random Sampling

Random sampling is crucial in data analysis as it ensures that every item or individual in a population has an equal chance of being selected. This is important because it helps in obtaining a representative sample that accurately reflects the whole group. In the given scenarios, random sampling is used to choose days for stock returns and to gather SAT scores from students.

There are several benefits of random sampling:

• Reduces Bias: Random selection minimizes bias, ensuring results can generally apply to the broader population.
• Represents the Entire Population: It targets a subset that reflects the diversity and characteristics of the entire group.
• Allows for Valid Statistical Analysis: With random samples, you can apply probability theory to infer results about the larger population.

Therefore, random sampling serves as a foundational method for diverse applications, from stock analysis to comparing student performance across different schools.

Stock Analysis

Stock analysis often involves looking at data points to understand and predict stock market behavior. In scenario (a), the approach was to determine if the returns from Intel and Southwest Airlines' stocks showed any relationship or similarity.

For a more in-depth analysis in stock markets, consider these common techniques:

• Historical Performance: Analyzing past data to make predictions about future performance.
• Technical Indicators: Employ metrics like moving averages and trend lines to understand market momentum.
• Fundamental Analysis: Examines financial statements to assess stock valuation.

While scenario (a) used random sampling for its initial exploration, the fact that the samples for Intel and Southwest were taken independently makes it a non-paired data comparison. This means that while both sets represent average performances over time, they do not reflect any direct connection or correlation between the two stocks.

SAT Score Comparison

Comparing SAT scores between different high schools can help understand educational outcomes. In the third scenario, SAT scores were sampled from two schools, aiming to compare the average results.

Here is what to consider when comparing SAT scores:

• Independent Samples: Each student's score is independent from the others, meaning results reflect each school's performance on its own.
• Average Score Differences: Focus on the average score in each school to determine which may perform better.
• Consider Other Factors: Beyond scores, consider school's resources, student backgrounds, and teaching quality.

Since each sample was collected independently, the data points were not paired. This method helps in providing a broad overview but lacks specific one-to-one comparisons.

Price Comparison

When looking at price comparison, as seen in scenario (b), the data sampling at Target and Walmart creates paired data. Prices for identical items are gathered from both retailers, establishing a direct one-to-one relationship for analysis.

Here’s why paired data is significant in price comparison:

• Direct Item Comparison: Prices are compared directly between stores, allowing for detailed price analysis on an item-by-item basis.
• Paired T-Test Usage: Since items are paired, a paired t-test can be used to determine significant differences in pricing strategies.
• Insight into Pricing Strategies: Helps to understand competitive pricing and potential consumer cost savings.

Thus, paired data allows for a more exact examination of price differences, offering clear insights into comparative price fluctuations between stores.