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Chapter 9: Problem 12

A simple random sample of size \(n<30\) for \(a\) quantitative variable has been obtained. Using the normal probability plot, the correlation between the variable and expected z-score, and the boxplot, judge whether a t-interval should be constructed. $$ n=9 ; \text { Correlation }=0.997 $$

Short Answer

Expert verified
Yes, a t-interval should be constructed due to the high correlation (0.997) indicating normality.

Step by step solution

01

- Understand the given data

Identify the sample size and the correlation given in the problem. Here, the sample size is 9 and the correlation is 0.997.
02

- Check the sample size condition

Since the sample size is less than 30 (<30), the t-interval can be used if the data appears to be approximately normally distributed.
03

- Analyze the correlation coefficient

The correlation between the variable and expected z-scores is 0.997, which is very close to 1. This strong correlation suggests that the data follows a normal distribution.
04

- Interpret the normal probability plot

A high correlation in the normal probability plot, such as 0.997, indicates that the data points lie close to a straight line, supporting normality.
05

- Consider the boxplot

Although the boxplot is not directly provided, a strong correlation from the normal probability plot already indicates the data is close to normal. A boxplot with mild skewness or outliers will not strongly affect this interpretation.
06

- Decision

Given the small sample size and the very high correlation (0.997) from the normal probability plot, it is appropriate to use the t-interval for constructing the confidence interval.

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Key Concepts

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

Simple Random Sample
A simple random sample is crucial for unbiased results in statistics. In this approach, every member of the population has an equal chance of being selected. This eliminates biases from the process. For the exercise, we have a simple random sample with a size of 9. This small sample size requires special methods, like checking normality, before proceeding with statistical analysis. It's fundamental because it ensures that the sample truly represents the population.
Normal Probability Plot
A Normal Probability Plot is a graphical technique for assessing whether or not a data set is approximately normally distributed. In this plot, observed data points are plotted against the expected z-scores of a normal distribution. If data points follow a straight line, the data is considered normal. In the current exercise, the plot shows a high correlation of 0.997. This close-to-one value suggests that our small sample closely follows a normal distribution, allowing us to make further statistical inferences.
Correlation Coefficient
The correlation coefficient measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1. A value closer to 1 or -1 indicates strong linear relationship, while a value near 0 indicates weak or no linear relationship. For our exercise, the correlation coefficient is 0.997. This high value indicates a very strong positive linear relationship between our data and the expected z-scores. This strong relationship supports the use of a normal distribution model for further analysis, even with a small sample size.
Boxplot
A Boxplot is a graphical representation that displays the distribution of a data set based on a five-number summary: minimum, first quartile, median, third quartile, and maximum. It also highlights outliers. Although not directly shown in the exercise, its utility lies in spotting deviations from normality, such as skewness or outliers. These characteristics can challenge the assumptions needed for statistical methods like t-intervals. Given our high correlation in the normal probability plot, we infer that our data is close to normal, even if the boxplot may show minor deviations.
Sample Size
Sample size is critical in statistical analysis. It affects the reliability and validity of the results. In general, larger sample sizes yield more accurate results. However, in this exercise, the sample size is 9, which is considered small. Small sample sizes make it crucial to test for normality using methods like the normal probability plot. Despite our small sample of 9, the high correlation of 0.997 allows us to proceed with constructing a t-interval. This ensures that confidence intervals computed are reliable and representative of the population.

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Most popular questions from this chapter

Certain statistics are difficult to bootstrap. One such statistic is the median. Consider the following to see why. (a) Simulate obtaining a random sample of 12 IQ scores. Recall IQ scores are approximately normally distributed with mean 100 and standard deviation \(15 .\) (b) Given that IQ scores are normally distributed, what is the median IQ score? (c) Obtain 1000 bootstrap samples from the data in part (a). Find the median of each bootstrap sample. (d) Draw a histogram of the bootstrap medians from part (c). What do you notice about the distribution? Find the \(95 \%\) confidence interval based on the 1000 bootstrap medians using the percentile method. (e) Repeat parts (a) through (d) using a random sample of 13 IQ scores. (f) Conclude that finding confidence intervals for medians is best if done where the sample size is even.

The exponential probability distribution can be used to model waiting time in line or the lifetime of electronic components. Its density function is skewed right. Suppose the wait-time in a line can be modeled by the exponential distribution with \(\mu=\sigma=5\) minutes. (a) Use StatCrunch, Minitab, or some other statistical software to generate 100 random samples of size \(n=4\) from this population. (b) Construct \(95 \% t\) -intervals for each of the 100 samples found in part (a). (c) How many of the intervals do you expect to include the population mean? How many of the intervals actually contain the population mean? Explain what your results mean. (d) Repeat parts (a)-(c) for samples of size \(n=15\) and \(n=25\). Explain what your results mean.

In a Gallup poll, \(64 \%\) of the people polled answered yes to the following question: "Are you in favor of the death penalty for a person convicted of murder?" The margin of error in the poll was \(3 \%,\) and the estimate was made with \(95 \%\) confidence. At least how many people were surveyed?

Construct the appropriate confidence interval. A simple random sample of size \(n=25\) is drawn from a population that is normally distributed. The sample variance is found to be \(s^{2}=3.97\). Construct a \(95 \%\) confidence interval for the population standard deviation.

Clayton Kershaw of the Los Angeles Dodgers is one of the premier pitchers in baseball. His most popular pitch is a four-seam fastball. The data in the next column represent the pitch speed (in miles per hour) for a random sample of 18 of his four-seam fastball pitches. $$ \begin{array}{llllll} \hline 93.63 & 93.83 & 94.18 & 94.71 & 95.52 & 95.07 \\ \hline 95.12 & 95.35 & 94.15 & 94.62 & 96.08 & 93.86 \\ \hline 94.75 & 94.70 & 95.28 & 95.49 & 95.77 & 93.34 \\ \hline \end{array} $$ (a) Is "pitch speed" a quantitative or qualitative variable? Why is it important to know this when determining the type of confidence interval you may construct? (b) Draw a normal probability plot to verify that "pitch speed" could come from a population that is normally distributed. (c) Draw a boxplot to verify the data set has no outliers. (d) Are the requirements for constructing a confidence interval for the mean pitch speed of a Clayton Kershaw four-seam fastball satisfied? (e) Construct and interpret a \(95 \%\) confidence interval for the mean pitch speed of a Clayton Kershaw four-seam fastball. (f) Do you believe that a \(95 \%\) confidence interval for the mean pitch speed of all major league pitchers' four-seam fastbal would be narrower or wider? Why?
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