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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

Number of Text Messages a Day A random sample of \(n=755\) US cell phone users age 18 and older in May 2011 found that the average number of text messages sent or received per day is 41.5 messages, 32 with standard error about \(6.1 .\) (a) State the population and parameter of interest. Use the information from the sample to give the best estimate of the population parameter. (b) Find and interpret a \(95 \%\) confidence interval for the mean number of text messages.

What Is an Average Budget for a Hollywood Movie? Data 2.7 on page 95 introduces the dataset HollywoodMovies, which contains information on more than 900 movies that came out of Hollywood between 2007 and \(2013 .\) We will consider this the population of all movies produced in Hollywood during this time period. (a) Find the mean and standard deviation for the budgets (in millions of dollars) of all Hollywood movies between 2007 and \(2013 .\) Use the correct notation with your answer. (b) Use StatKey or other technology to generate a sampling distribution for the sample mean of budgets of Hollywood movies during this period using a sample size of \(n=20\). Give the shape and center of the sampling distribution and give the standard error.

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.

Automobile Depreciation For a random sample of 20 automobile models, we record the value of the model as a new car and the value after the car has been purchased and driven 10 miles. \({ }^{47}\) The difference between these two values is a measure of the depreciation on the car just by driving it off the lot. Depreciation values from our sample of 20 automobile models can be found in the dataset CarDepreciation. (a) Find the mean and standard deviation of the Depreciation amounts in CarDepreciation. (b) Use StatKey or other technology to create a bootstrap distribution of the sample mean of depreciations. Describe the shape, center, and spread of this distribution. (c) Use the standard error obtained in your bootstrap distribution to find and interpret a \(95 \%\) confidence interval for the mean amount a new car depreciates by driving it off the lot.

Mean number of cell phone calls made or received per day by cell phone users. In a survey of 1917 cell phone users, the mean was 13.10 phone calls a day.
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