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Chapter 2: Problem 89

Find and interpret the \(z\) -score for the data value given. The value 5.2 in a dataset with mean 12 and standard deviation 2.3

Short Answer

Expert verified
The z-score for the value 5.2 in the given dataset is approximately -2.9565, indicating that it is approximately 2.9565 standard deviations below the mean.

Step by step solution

01

Understand the Z-Score Formula

The formula for calculating the z-score is: z = (X - 渭) / 蟽. In this formula, z is the z-score, X is the value from the dataset, 渭 is the mean of the dataset, and 蟽 is the standard deviation of the dataset.
02

Insert Given Values into the Z-Score Formula

Now that the Z-score formula is known, insert the given values into the equation: z = (5.2 - 12) / 2.3.
03

Calculate the Z-Score

Compute the given values to get the Z-score. This will result in a negative value, as 5.2 is less than the mean 12. Upon performing the calculation, we arrive at z approximately equal to -2.9565.
04

Interpret the Z-Score

The z-score indicates how many standard deviations an element is from the mean. So, a z-score of -2.9565 means that the value 5.2 is approximately 2.9565 standard deviations below the mean in this dataset.

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Key Concepts

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

Mean and Standard Deviation
Two fundamental concepts in statistics are the mean and the standard deviation. These measures help us understand data distributions.
  • Mean represents the average value of a dataset. It is found by summing all the data points and dividing by the number of data points. For example, if your data points are 5, 10, and 15, the mean would be (5 + 10 + 15) / 3 = 10.
  • Standard Deviation measures the spread of the data points around the mean. A low standard deviation means the data points are close to the mean, while a high standard deviation indicates the data points are spread out over a larger range.
In our exercise, the mean is 12, and the standard deviation is 2.3. Understanding these allows us to determine how typical or atypical a particular data point is.
Statistical Interpretation
Statistical interpretation involves making sense of data by using statistical measures to comprehend and communicate the data's significance. A key tool here is the z-score.
  • Z-Score: It measures how many standard deviations a particular value (from the dataset) is away from the mean. For instance, a z-score of 0 suggests the value is equal to the mean.
  • In our exercise, the calculation gave us a z-score of approximately -2.9565. This negative sign indicates the value in question is less than the mean.
The z-score informs you about the relative position of a data value within a dataset. A z-score of -2.9565 shows that the value of 5.2 falls below the mean, specifically about 2.96 standard deviations below.
Dataset Analysis
Analyzing a dataset involves examining its components, such as mean, standard deviation, and z-scores, to draw insights.
  • Comparisons Using Z-Scores: Z-scores allow us to easily compare different data points within a dataset and even across different datasets.
  • Given the z-score of approximately -2.9565 for the value 5.2, one can conclude this data point is considerably lower than the dataset's mean. This type of analysis helps identify outliers or anomalies.
This example highlights the importance of statistical tools in understanding the characteristics and behaviors of data within a dataset.

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Most popular questions from this chapter

Each describe a sample. The information given includes the five number summary, the sample size, and the largest and smallest data values in the tails of the distribution. In each case: (a) Clearly identify any outliers. (b) Draw a boxplot. Five number summary: (5,10,12,16,30)\(;\) \(n=40\) Tails: \(5,5,6,6,6, \ldots, 22,22,23,28,30\)

For each set of data in Exercises 2.43 to 2.46: (a) Find the mean \(\bar{x}\). (b) Find the median \(m\). (c) Indicate whether there appear to be any outliers. If so, what are they? \(\begin{array}{lllllll}& 41, & 53, & 38, & 32, & 115, & 47, & 50\end{array}\)

In Exercises 2.114 to \(2.117,\) sketch a curve showing a distribution that is symmetric and bell-shaped and has approximately the given mean and standard deviation. In each case, draw the curve on a horizontal axis with scale 0 to 10 . Mean 3 and standard deviation 1

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)

Laptop Computers and Sperm Count Stu dies have shown that heating the scrotum by jus \(1^{\circ} \mathrm{C}\) can reduce sperm count and sperm quality so men concerned about fertility are cautioned to avoid too much time in the hot tub or sauna. A new study \(^{41}\) suggests that men also keep their lap top computers off their laps. The study measurec scrotal temperature in 29 healthy male volunteer as they sat with legs together and a laptop compute on the lap. Temperature increase in the left scrotun over a 60 -minute session is given as \(2.31 \pm 0.96\) anc a note tells us that "Temperatures are given as \({ }^{\circ} \mathrm{C}\) values are shown as mean \(\pm \mathrm{SD} . "\) The abbreviatior SD stands for standard deviation. (Men who sit witl their legs together without a laptop computer do not show an increase in temperature.) (a) If we assume that the distribution of the temper ature increases for the 29 men is symmetric anc bell-shaped, find an interval that we expect to contain about \(95 \%\) of the temperature increases (b) Find and interpret the \(z\) -score for one of the men, who had a temperature increase of \(4.9^{\circ}\).
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