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Chapter 2: Problem 93

Use the \(95 \%\) rule and the fact that the summary statistics come from a distribution that is symmetric and bell-shaped to find an interval that is expected to contain about \(95 \%\) of the data values. A bell-shaped distribution with mean 1000 and standard deviation 10

Short Answer

Expert verified
The interval that contains approximately 95% of the data values is between 980 and 1020.

Step by step solution

01

Identify the mean and standard deviation

Amand the given statistics, the mean of the distribution is \(\mu = 1000\) and the standard deviation is \(\sigma = 10\)
02

Apply the Empirical Rule

From the empirical rule, we know that roughly 95% of the data in a symmetric, bell-shaped distribution falls within 2 standard deviations of the mean.
03

Compute the interval

To compute the interval that contains approximately 95% of the data, add and subtract two standard deviations to the mean. The lower bound is \(\mu - 2\sigma = 1000 - 2 * 10 = 980\), and the upper bound is \(\mu + 2\sigma = 1000 + 2 * 10 = 1020\)

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Key Concepts

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

Bell-Shaped Distribution
When we encounter the term bell-shaped distribution, we're talking about a graphical representation of data that is symmetric and unimodal, which means that it has a single peak, and tails off evenly on both sides of that peak. The highest point on this 'bell' corresponds with the mean of the data, which is also the median and mode in a perfectly symmetrical distribution.

The bell shape is indicative of many natural phenomena and human behaviors, making it particularly useful in various fields such as psychology, economics, and biology. In such distributions, most values cluster around a central region, with fewer instances as you move away in either direction from that center. This concept is central to many statistical operations and interpretations.
Standard Deviation
The term standard deviation is used to measure the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values are close to the mean of the set, while a high standard deviation shows that the values are spread out over a wider range.

To put it in perspective, if we observe a dataset with a smaller standard deviation, it means the data points tend to be very close to the mean, resulting in a steeper bell-shaped curve. Conversely, a larger standard deviation leads to a flatter curve. It's a critical tool in the field of statistics, as it provides insights into the uncertainty or 'spread' of a given dataset.
95% Confidence Interval
Now, the 95% confidence interval is a term that often induces confusion, but it's actually a straightforward concept. It refers to the range within which we can expect 95% of the data values to fall, based on the empirical rule. This rule tells us that for a bell-shaped distribution, about 68% of the data falls within one standard deviation of the mean, about 95% fall within two standard deviations, and about 99.7% within three.

In the context of our example, the interval between 980 and 1020 encompasses the middle 95% of the data. Finding this interval is invaluable for making predictions and understanding the variability of data. It offers a way to communicate the precision of our estimated mean; narrower intervals suggest greater precision. In many scientific fields, the use of a 95% confidence interval is standard for reporting the reliability of estimated values.

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Most popular questions from this chapter

Indicate whether the five number summary corresponds most likely to a distribution that is skewed to the left, skewed to the right, or symmetric. (0,15,22,24,27)

Indicate whether the five number summary corresponds most likely to a distribution that is skewed to the left, skewed to the right, or symmetric. (22.4,30.1,36.3,42.5,50.7)

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Of the students who identified their class year in the survey, 94 were first years, 195 were sophomores, 35 were juniors, and 36 were seniors. Find \(\hat{p}\), the proportion who are upperclass students (juniors or seniors.)

Apparently, sexual frustration increases the desire for alcohol, at least in fruit flies. Scientists \(^{33}\) randomly put 24 fruit flies into one of two situations. The 12 fruit flies in the "mating" group were allowed to mate freely with many available females eager to mate. The 12 in the "rejected" group were put with females that had already mated and thus rejected any courtship advances. After four days of either freely mating or constant rejection, the fruit flies spent three days with unlimited access to both normal fruit fly food and the same food soaked in alcohol. The percent of time each fly chose the alcoholic food was measured. The fruit flies that had freely mated chose the two types of food about equally often, choosing the alcohol variety on average \(47 \%\) of the time. The rejected males, however, showed a strong preference for the food soaked in alcohol, selecting it on average \(73 \%\) of the time. (The study was designed to study a chemical in the brain called neuropeptide that might play a role in addiction.) (a) Is this an experiment or an observational study? (b) What are the cases in this study? What are the variables? Which is the explanatory variable and which is the response variable? (c) We are interested in the difference in means, where the means measure the average percent preference for alcohol \((0.47\) and 0.73 in this case). Find the difference in means and give the correct notation for your answer, using the correct notation for a mean, subscripts to identify groups, and a minus sign. (d) Can we conclude that rejection increases a male fruit fly's desire for alcohol? Explain.
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