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Find and interpret the z-score for the data value given. The value 8.1 in a dataset with mean 5 and standard deviation 2

Short Answer

Expert verified
The z-score is 1.55, which means the data value 8.1 is 1.55 standard deviations above the mean of the dataset.

Step by step solution

01

Identify the given values in the problem

From the problem, the following values are given: Value = 8.1, Mean = 5, Standard deviation = 2
02

Apply the values to the Z-score formula

The Z-score formula is \(z = \frac{Value - Mean}{Standard deviation}\). Now, we substitute the given values into the formula. \(z = \frac{8.1 - 5}{2}\)
03

Calculate the Z-score

Performing the calculation in the numerator first, then dividing, we get: \(z = \frac{3.1}{2} = 1.55\)
04

Interpret the Z-score

A Z-score of 1.55 means the data value 8.1 is 1.55 standard deviations above the mean of the dataset

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Key Concepts

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

Standard Deviation
Understanding the standard deviation is crucial when interpreting data. It is a measure that tells us how much variation or dispersion there is from the average (mean) of a set of data.

Think of it like this: if we have a group of people and we measure their heights, the average height might tell us one part of the story, but knowing how much the heights vary tells us a lot more. If everyone is around the same height, we have a small standard deviation. If we have some really tall and some really short people, the standard deviation will be larger, signaling a wide range of differences. In the context of the exercise, a standard deviation of 2 means that the values in the dataset typically deviate from the mean by a distance of 2 units.
Data Value
The term 'data value' refers to a single piece of information that's been recorded. In our daily lives, we compare these values to make decisions. For instance, if you're looking at temperatures over a week to decide what to wear, each day's temperature is a data value.

In the exercise, the data value in question is 8.1. This is a specific observation or measurement from the dataset we are analyzing. It's like noticing on one particular day that the temperature spiked to 8.1 degrees above what's usual for this time of year, and you're trying to figure out how significant that spike is.
Mean
The mean, often referred to as the average, is a central value of a set of numbers. It's calculated by adding up all the values and then dividing by the number of values. In our previous analogy, it's like finding out the average temperature for the week.

The mean gives us a reference point to compare individual data values. In the exercise, the mean of the dataset is 5, which allows us to see that the value 8.1 is higher than what is typical for this set of data.
Z-score Formula
The Z-score formula is a very useful tool for statisticians. It's given as \( z = \frac{Value - Mean}{Standard\ deviation} \). The Z-score helps us describe where a certain data value lies in relation to the mean of the dataset, in terms of standard deviations.

Let's break down the calculation from the exercise. We have the data value (8.1), the mean (5), and the standard deviation (2). By substituting these into the formula, \( z = \frac{8.1 - 5}{2} \), we find that the data value 8.1 has a Z-score of 1.55.

What Does the Z-score Tell Us?

It tells us that 8.1 is 1.55 standard deviations above the mean. This is quite a bit higher than the average, signaling that 8.1 is not just a random variation but could be significant. If you imagine a bell curve, 8.1 would be sitting to the right of the center.

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Most popular questions from this chapter

2Does Sexual Frustration Increase the Desire for Alcohol? Apparently, sexual frustration increases the desire for alcohol, at least in fruit flies. Scientists \(^{35}\) 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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