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Chapter 9: Problem 484

What can be said about a sample of observations whose standard deviation is zero?

Short Answer

Expert verified
A sample of observations with a standard deviation of zero signifies that all the observations have the same value, i.e., equal to the mean. This implies that there is no spread or variability among the observations, as they all share the same value.

Step by step solution

01

Definition of Standard Deviation

Standard deviation is a quantity which measures the dispersion or spread of a set of data points around the mean. It is calculated using the following formula: \(s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}\) where: - \(s\) is the sample standard deviation - \(\bar{x}\) is the sample mean (average) - \(x_i\) are the individual data points in the sample - \(n\) is the number of data points in the sample
02

Interpret Standard Deviation Value

When the standard deviation is zero, this implies that the quantity \(\sum (x_i - \bar{x})^2\) equal to zero as well. This can be written as: \(\sum (x_i - \bar{x})^2 = 0\)
03

Analyze \(\sum (x_i - \bar{x})^2 = 0\)

The only way for this equation to hold true is if each term in the sum is equal to zero, meaning that: \((x_i - \bar{x})^2 = 0\) for all \(i\) Since squaring a number cannot make it negative, it means that each term \((x_i - \bar{x})\) must be zero: \((x_i - \bar{x}) = 0\)
04

Observations Have the Same Value

From the previous step, we can deduce that: \(x_i = \bar{x}\) for all \(i\) This means that every data point in the sample has the same value as the mean, i.e., all observations in the sample have the same value.
05

Conclusion

So, to answer the question, a sample of observations whose standard deviation is zero are all equal to each other. There is no spread or variability among the observations, as they all share the same value.

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Most popular questions from this chapter

Find the variance of the random variable \(\mathrm{X}+\mathrm{b}\) where \(\mathrm{X}\) has variance, \(\operatorname{Var} \mathrm{X}\) and \(\mathrm{b}\) is a constant.

Let \(\mathrm{X}\) be a normally distributed random variable representing the hourly wage in a certain craft. The mean of the hourly wage is \(\$ 4.25\) and the standard deviation is \(\$ .75 .\) (a) What percentage of workers receive hourly wages between \(\$ 3.50\) and \(\$ 4.90 ?\) (b) What hourly wage represents the 95 th percentile?

Suppose we have a binomial distribution for which \(\mathrm{H}_{0}\) is \(\mathrm{p}=1 / 2\) where \(\mathrm{p}\) is the probability of success on a single trial. Suppose the type I error, \(\alpha=.05\) and \(\mathrm{n}=100 .\) Calculate the power of this test for each of the following alternate hypotheses, \(\mathrm{H}_{1}: \mathrm{p}=.55, \mathrm{p}=.60, \mathrm{p}=.65, \mathrm{p}=.70\), and \(\mathrm{p}=.75 .\) Do the same when \(\alpha=.01\).

Given the probability distribution of the random variable \(\mathrm{X}\) in the table below, compute \(\mathrm{E}(\mathrm{X})\) and \(\operatorname{Var}(\mathrm{X})\). $$ \begin{array}{|c|c|} \hline \mathrm{x}_{\mathrm{i}} & \operatorname{Pr}\left(\mathrm{X}=\mathrm{x}_{i}\right) \\ \hline 0 & 8 / 27 \\ \hline 1 & 12 / 27 \\ \hline 2 & 6 / 27 \\ \hline 3 & 1 / 27 \\ \hline \end{array} $$

The seven dwarfs challenged the Harlem Globetrotters to a basketball game. Besides the obvious difference in height, we are interested in constructing a \(95 \%\) confidence in ages between dwarfs and basketball players. The respective ages are, $$ \begin{array}{|l|l|l|l|} \hline \text { Dwardfs } & & \text { Globetrotters } & \\ \hline \text { Sneezy } & 20 & \text { Meadowlark } & 43 \\ \hline \text { Grumpy } & 39 & \text { Curley } & 37 \\ \hline \text { Dopey } & 23 & \text { Marques } & 45 \\ \hline \text { Doc } & 41 & \text { Bobby Joe } & 25 \\ \hline \text { Sleepy } & 35 & \text { Theodis } & 34 \\ \hline \text { Happy } & 29 & & \\ \hline \text { Bashful } & 31 & & \\ \hline \end{array} $$ Can you construct the interval? Assume the variance in age is the same for dwarfs and Globetrotters.
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