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Chapter 3: Problem 53

In Exercises 3.51 to 3.56 , information about a sample is given. Assuming that the sampling distribution is symmetric and bell-shaped, use the information to give a \(95 \%\) confidence interval, and indicate the parameter being estimated. $$ r=0.34 \text { and the standard error is } 0.02 \text { . } $$

Short Answer

Expert verified
The \(95\%\) confidence interval for the population correlation coefficient 'r' is \(0.302, 0.378\).

Step by step solution

01

Find the Z-score

First, we need to find the Z-score that corresponds to a \(95\%\) confidence level. Since we want the middle \(95\%\) , we are left with \(5\%\) in the two tails of the Normal Distribution. Half of \(5\%\) is \(2.5\%\) in each tail. So, we look up \(2.5\%\) in the Z-table or use a calculator to find the Z-score. The Z-score for \(95\%\) confidence is roughly \(\pm 1.96\).
02

Insert into the formula

Now we have all the values to insert into our confidence interval formula \(X \pm Z * SE\). Here, \(X = 0.34\), which is the given sample statistic, \(Z = 1.96\), and \(SE = 0.02\). Substitute these values into the formula to get \(0.34 \pm 1.96 * 0.02\).
03

Calculate the confidence interval

Calculate the upper and lower limit of the confidence interval by performing the operations. The lower limit is \(0.34 - 1.96 * 0.02 = 0.302\), and the upper limit is \(0.34 + 1.96 * 0.02 = 0.378\).
04

Interpret the result

The \(95\%\) confidence interval is \(0.302, 0.378\). This means that we're \(95\%\) confident that the actual population parameter lies between these two numbers. The parameter being estimated here is 'r', the population correlation coefficient.

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Key Concepts

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

Z-Score
A Z-score is a way to put your data into perspective. It's a numerical measurement that describes a value's relationship to the mean of a group of values. When working with confidence intervals, the Z-score helps us understand how likely a result is to occur under a normal distribution.

For example, when determining a 95% confidence interval, you're looking for the Z-score that leaves 2.5% in each tail of the distribution (since 5% is outside the middle 95%). This value is approximately 1.96.
  • Z-scores are found using a Z-table or calculator.
  • They help determine the confidence interval limits.
Knowing how to find and work with Z-scores is crucial for estimating where the true population parameter might lie.
Sampling Distribution
A sampling distribution refers to the distribution of a given statistic based on a random sample.

When you take a sample and calculate a statistic, like the mean or proportion, you're working within a sampling distribution. Since your sample is likely to differ from another sample, the sampling distribution captures possible variations across different samples.
  • A sampling distribution is usually bell-shaped and symmetric.
  • It allows us to understand how sample statistics vary instead of focusing on a single outcome.
In the context of the exercise, this concept lets us apply normal distribution concepts to real-world data.
Population Parameter Estimation
Estimating a population parameter involves using sample data to calculate an interval believed to contain the true population parameter.

In the exercise, the parameter being estimated is 'r', the population correlation coefficient. Through the calculated confidence interval, we assume that the true correlation coefficient falls between the identified limits with a certain level of confidence (e.g., 95%).
  • We use sample statistics to estimate unknown population parameters.
  • Confidence intervals provide a range where the population parameter is likely to lie.
Accurate parameter estimation helps make inferences about the larger population based on our sample data.

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Most popular questions from this chapter

Socially Conscious Consumers In March 2015, a Nielsen global online survey "found that consumers are increasingly willing to pay more for socially responsible products."11 Over 30,000 people in 60 countries were polled about their purchasing habits, and \(66 \%\) of respondents said that they were willing to pay more for products and services from companies who are committed to positive social and environmental impact. We are interested in estimating the proportion of all consumers willing to pay more. Give notation for the quantity we are estimating, notation for the quantity we are using to make the estimate, and the value of the best estimate. Be sure to clearly define any parameters in the context of this situation.

Playing Video Games A new study provides some evidence that playing action video games strengthens a person's ability to translate sensory information quickly into accurate decisions. Researchers had 23 male volunteers with an average age of 20 look at moving arrays on a computer screen and indicate the direction in which the dots were moving \(^{33}\) Half of the volunteers ( 11 men) reported playing action video games at least five times a week for the previous year, while the other 12 reported no video game playing in the previous year. The response time and the accuracy score were both measured. A \(95 \%\) confidence interval for the mean response time for game players minus the mean response time for non-players is -1.8 to -1.2 seconds, while a \(95 \%\) confidence interval for mean accuracy score for game players minus mean accuracy score for non-players is -4.2 to +5.8 . (a) Interpret the meaning of the \(95 \%\) confidence interval for difference in mean response time. (b) Is it plausible that game players and non-game players are basically the same in response time? Why or why not? If not, which group is faster (with a smaller response time)? (c) Interpret the meaning of the \(95 \%\) confidence interval for difference in mean accuracy score. (d) Is it plausible that game players and non-game players are basically the same in accuracy? Why or whynot? If not, which group is more accurate?

To create a confidence interval from a bootstrap distribution using percentiles, we keep the middle values and chop off some number of the lowest values and the highest values. If our bootstrap distribution contains values for 1000 bootstrap samples, indicate how many we chop off at each end for each confidence level given. (a) \(95 \%\) (b) \(90 \%\) (c) \(98 \%\) (d) \(99 \%\)

Daily Tip Revenue for a Waitress Data 2.12 on page 123 describes information from a sample of 157 restaurant bills collected at the First Crush bistro. The data is available in RestaurantTips. Two intervals are given below for the average tip left at a restaurant; one is a \(90 \%\) confidence interval and one is a \(99 \%\) confidence interval. Interval A: 3.55 to 4.15 Interval B: 3.35 to 4.35 (a) Which one is the \(90 \%\) confidence interval? Which one is the \(99 \%\) confidence interval? (b) One waitress generally waits on 20 tables in an average shift. Give a range for her expected daily tip revenue, using both \(90 \%\) and \(99 \%\) confidence. Interpret your results.

Bisphenol A in Your Soup Cans Bisphenol A (BPA) is in the lining of most canned goods, and recent studies have shown a positive association between BPA exposure and behavior and health problems. How much does canned soup consumption increase urinary BPA concentration? That was the question addressed in a recent study \(^{34}\) in which consumption of canned soup over five days was associated with a more than \(1000 \%\) increase in urinary BPA. In the study, 75 participants ate either canned soup or fresh soup for lunch for five days. On the fifth day, urinary BPA levels were measured. After a two-day break, the participants switched groups and repeated the process. The difference in BPA levels between the two treatments was measured for each participant. The study reports that a \(95 \%\) confidence interval for the difference in means (canned minus fresh) is 19.6 to \(25.5 \mu \mathrm{g} / \mathrm{L}\). (a) Is this a randomized comparative experiment or a matched pairs experiment? Why might this type of experiment have been used? (b) What parameter are we estimating? (c) Interpret the confidence interval in terms of BPA concentrations. (d) If the study had included 500 participants instead of \(75,\) would you expect the confidence interval to be wider or narrower?
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