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Ch. 10 - Correlation and Regression
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 10 - Correlation and RegressionProblem 10.1.34
Chapter 10, Problem 10.1.34

Randomization
For Exercises 33–36, repeat the indicated exercise using the resampling method of randomization.
Powerball Jackpots and Tickets Sold Exercise 14

Verified step by step guidance
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Identify the problem: The task involves using the resampling method of randomization to analyze the relationship between Powerball jackpots and tickets sold. Resampling is a statistical method that involves repeatedly drawing samples from observed data to assess variability or test hypotheses.
Step 1: Collect the data. Gather the observed data for Powerball jackpots and the corresponding number of tickets sold. Ensure the data is organized in pairs, where each pair represents a jackpot amount and the number of tickets sold for that jackpot.
Step 2: Define the null hypothesis. For example, the null hypothesis might state that there is no relationship between the size of the jackpot and the number of tickets sold. This will guide the randomization process.
Step 3: Randomize the data. Shuffle the observed data for the number of tickets sold while keeping the jackpot amounts fixed. This breaks any existing relationship between the two variables, simulating the null hypothesis.
Step 4: Calculate the test statistic for the randomized data. For example, compute the correlation coefficient or another measure of association between the jackpot amounts and the randomized ticket sales data. Repeat this process many times (e.g., 1,000 or more) to create a distribution of the test statistic under the null hypothesis.
Step 5: Compare the observed test statistic to the randomization distribution. Determine the p-value by finding the proportion of randomized test statistics that are as extreme or more extreme than the observed test statistic. Use this p-value to decide whether to reject or fail to reject the null hypothesis.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Randomization

Randomization is a statistical technique used to assign subjects to different groups in a way that eliminates bias. This method ensures that each participant has an equal chance of being assigned to any group, which helps in making the results more reliable and generalizable. In the context of experiments, randomization helps in controlling for confounding variables, thereby enhancing the validity of the conclusions drawn.
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Resampling Method

The resampling method involves repeatedly drawing samples from a dataset and calculating a statistic of interest for each sample. This technique, which includes methods like bootstrapping and permutation tests, allows statisticians to estimate the distribution of a statistic without making strong parametric assumptions. It is particularly useful for assessing the variability of an estimate and for hypothesis testing in situations where traditional methods may not be applicable.
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Powerball Jackpots and Tickets Sold

The Powerball lottery is a game where players select numbers in hopes of winning a jackpot, which is determined by the number of tickets sold and the total prize pool. Analyzing the relationship between jackpots and tickets sold involves understanding probability and expected value, as well as how these factors influence player behavior and overall lottery revenue. This analysis can provide insights into the effectiveness of marketing strategies and the economic impact of lottery games.
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Related Practice
Textbook Question

Finding the Best Model

In Exercises 5–16, construct a scatterplot and identify the mathematical model that best fits the given data. Assume that the model is to be used only for the scope of the given data, and consider only linear, quadratic, logarithmic, exponential, and power models.

Richter Scale The table lists different amounts (metric tons) of the explosive TNT and the corresponding value measured on the Richter scale resulting from explosions of the TNT.

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Textbook Question

Interpreting a Computer Display

In Exercises 9–12, refer to the display obtained by using the paired data consisting of weights (pounds) and highway fuel consumption amounts (mi/gal) of the large cars included in Data Set 35 “Car Data” in Appendix B. Along with the paired weights and fuel consumption amounts, StatCrunch was also given the value of 4000 pounds to be used for predicting highway fuel consumption.



Testing for Correlation Use the information provided in the display to determine the value of the linear correlation coefficient. Is there sufficient evidence to support a claim of a linear correlation between weights of large cars and the highway fuel consumption amounts?

Textbook Question

Interpreting a Computer Display

In Exercises 9–12, refer to the display obtained by using the paired data consisting of weights (pounds) and highway fuel consumption amounts (mi/gal) of the large cars included in Data Set 35 “Car Data” in Appendix B. Along with the paired weights and fuel consumption amounts, StatCrunch was also given the value of 4000 pounds to be used for predicting highway fuel consumption.


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Predicting Highway Fuel Consumption Using a car weight of x = 4000 (pounds), what is the single value that is the best predicted amount of highway fuel consumption?

Textbook Question

Finding the Best Model

In Exercises 5–16, construct a scatterplot and identify the mathematical model that best fits the given data. Assume that the model is to be used only for the scope of the given data, and consider only linear, quadratic, logarithmic, exponential, and power models.

Sound Intensity The table lists intensities of sounds as multiples of a basic reference sound. A scale similar to the decibel scale is used to measure the sound intensity.

Textbook Question

Coefficient of Determination Using the heights and weights described in Exercise 1, the linear correlation coefficient r is 0.394. Find the value of the coefficient of determination. What practical information does the coefficient of determination provide?

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Textbook Question

Standard Error of Estimate A random sample of 118 different female statistics students is obtained and their weights are measured in kilograms and in pounds. Using the 118 paired weights (weight in kg, weight in lb), what is the value of se? For a female statistics student who weighs 100 lb, the predicted weight in kilograms is 45.4 kg. What is the 95% prediction interval?