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Chapter 3: Problem 47

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. $$ r=0.34 \text { and the standard error is } 0.02 . $$

Short Answer

Expert verified
The \(95\%\) confidence interval is found by subtracting and adding \(1.96\) times the standard error from/to the sample parameter estimate, thus the interval is \([0.34 - 1.96*0.02, 0.34 + 1.96*0.02]\). It is estimating the population correlation coefficient.

Step by step solution

01

Identify given values

Here, the sample parameter estimate \(r\) is given as 0.34, and the standard error is given as 0.02.
02

Compute Confidence Interval

Since the \(95\%\) confidence interval corresponds to \(卤1.96\) standard deviations from the mean in a normal distribution, we can construct the confidence interval by multiplying the standard error by 1.96, and adding/subtracting this value from the sample parameter estimate. Mathematically, it can be represented as follows: \[ \text{Confidence Interval} = r 卤 1.96 \times \text{Standard Error} \] Thus, substitute the given values into the formula, we get: \[ \text{Confidence Interval} = 0.34 卤 1.96 \times 0.02 \] Calculate the above expression to find the \(95\%\) confidence interval.
03

Interpret the Result

The calculated confidence interval will provide the range of likely values for the parameter being estimated. Remember, a \(95\%\) confidence interval means that there is a \(95\%\) probability that the interval will contain the true parameter value.

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Key Concepts

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

Correlation Coefficient
The correlation coefficient, often denoted as \( r \), is a powerful statistical tool. It measures the strength and direction of a linear relationship between two variables. Its value ranges between -1 and 1, where:
  • 0 indicates no correlation
  • 1 indicates a perfect positive correlation
  • -1 indicates a perfect negative correlation
A positive \( r \) value, like the given 0.34, suggests a moderate positive relationship, meaning as one variable increases, the other tends to increase as well. Understanding this helps in predicting trends and insights about the interdependence of variables. In practical scenarios, it鈥檚 vital to interpret the correlation in the context of the data, watching out for outliers that could skew the results.
Standard Error
Standard Error (SE) is crucial for understanding the precision of a sample statistic as an estimate of a population parameter. It essentially represents the variability of a sample statistic, like the correlation coefficient, from the true population parameter.
  • The lower the SE, the more precision in the estimate of the population parameter.
  • The formula to compute SE often involves dividing the sample standard deviation by the square root of the sample size.
For the correlation coefficient, a standard error of 0.02 indicates that we can be confident in the stability of our sample estimate, which is critical in forming a reliable confidence interval. The SE directly influences the width of the confidence interval, making it narrower when the SE is smaller.
Normal Distribution
The concept of normal distribution is central to statistics, often depicted as a symmetric, bell-shaped curve. It鈥檚 the foundation for many statistical methods, including confidence intervals. A normal distribution exhibits the following characteristics:
  • The mean divides the curve into two equal halves.
  • The standard deviation determines the width of the curve; about 68% of data falls within one standard deviation.
  • Around 95% of data is within two standard deviations.
In this exercise, understanding the normal distribution helps us know that a 95% confidence interval can be approximated using \( \pm 1.96 \) times the standard deviation (or standard error). This statistical property allows us to estimate the range within which the true population parameter is likely to lie, based on the sample data.

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Most popular questions from this chapter

Laptop Computers A survey conducted in May of 2010 asked 2252 adults in the US "Do you own a laptop computer?" The number saying yes was 1238 . What is the best estimate for the proportion of US adults owning a laptop computer? 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.

Gender in the Rock and Roll Hall of Fame From its founding through \(2012,\) the Rock and Roll Hall of Fame has inducted 273 groups or individuals. Forty-one of the inductees have been female or have included female members. \({ }^{16}\) The full dataset is available in RockandRoll. (a) What proportion of inductees have been female or have included female members? Use the correct notation with your answer. (b) If we took many samples of size 50 from the population of all inductees and recorded the proportion female or with female members for each sample, what shape do we expect the distribution of sample proportions to have? Where do we expect it to be centered?

What Is the Effect of Including Some Indifferent Fish? In the experiment described above under Fish Democracies, the schools of fish in the study with an opinionated minority and a less passionate majority picked the majority option only about \(17 \%\) of the time. However, when groups also included 10 fish with no opinion, the schools of fish picked the majority option \(61 \%\) of the time. We want to estimate the effect of adding the fish with no opinion to the group, which means we want to estimate the difference in the two proportions. We learn from the study that the standard error for estimating this difference is about \(0.14 .\) Define the parameter we are estimating, give the best point estimate, and find and interpret a \(95 \%\) confidence interval. Is it plausible that adding indifferent fish really has no effect on the outcome?

In estimating the mean score on a fitness exam, we use an original sample of size \(n=30\) and a bootstrap distribution containing 5000 bootstrap samples to obtain a \(95 \%\) confidence interval of 67 to 73 . In Exercises 3.90 to 3.95 , a change in this process is described. If all else stays the same, which of the following confidence intervals \((A, B,\) or \(C)\) is the most likely result after the change: \(\begin{array}{lll}A .66 \text { to } 74 & B .67 \text { to } 73 & \text { C. } 67.5 \text { to } 72.5\end{array}\) Using an original sample of size \(n=45\)

Small Sample Size and Outliers As we have seen, bootstrap distributions are generally symmetric and bell-shaped and centered at the value of the original sample statistic. However, strange things can happen when the sample size is small and there is an outlier present. Use StatKey or other technology to create a bootstrap distribution for the standard deviation based on the following data: \(\begin{array}{llllll}8 & 10 & 7 & 12 & 13 & 8\end{array}\) \(\begin{array}{ll}10 & 50\end{array}\) Describe the shape of the distribution. Is it appropriate to construct a confidence interval from this distribution? Explain why the distribution might have the shape it does.
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