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Chapter 7: Problem 52

Multiple choice: Standard deviation Which of the following is not correct? The standard deviation of a statistic describes a. The standard deviation of the sampling distribution of that statistic. b. The standard deviation of the sample data measurements. c. How close that statistic falls to the parameter that it estimates. d. The variability in the values of the statistic for repeated random samples of size \(n\).

Short Answer

Expert verified
Option c is not correct.

Step by step solution

01

Understand the Concept of Standard Deviation

The standard deviation is a measure of how spread out numbers are in a data set. It's used to quantify the amount of variation or dispersion in a set of data values.
02

Analyze Option a

Option a states that the standard deviation of a statistic describes the standard deviation of the sampling distribution of that statistic. This is correct because the standard deviation measures the spread of the sampling distribution and is often referred to as the standard error.
03

Analyze Option b

Option b states that the standard deviation describes the standard deviation of the sample data measurements. This is also correct, as the standard deviation of the sample calculates the variability of the direct sample data.
04

Analyze Option c

Option c states that the standard deviation describes how close that statistic falls to the parameter that it estimates. This is incorrect as standard deviation measures the spread of data, not the accuracy or closeness of statistic to parameter. Instead, the confidence interval would be more related to estimating how close a statistic is to a parameter.
05

Analyze Option d

Option d states that the standard deviation describes the variability in the values of the statistic for repeated random samples of size \(n\). This is correct because the standard deviation can measure how much those statistics vary from each other across different samples.

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Key Concepts

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

Sampling Distribution
A sampling distribution is a vital concept in statistics that relates to the distribution of a statistic over many samples. When you draw a sample from a population, you can calculate a statistic, such as the mean or the proportion. If you were to take multiple samples and calculate the same statistic for each sample, you would get a distribution of those statistics. This distribution is known as the sampling distribution.

Error! Self-referencing characteristics include:
  • Centered around the true parameter of the population, providing an estimate of where the true statistic might lie.
  • With larger sample sizes, the sampling distribution tends to be normally distributed, regardless of the shape of the population distribution, thanks to the Central Limit Theorem.
  • The mean of the sampling distribution of the sample mean equals the mean of the population.
By studying sampling distributions, statisticians can make inferences about population parameters and understand the precision of sample statistics.
Sample Data
Sample data refers to the subset of data collected from a larger population for the purpose of making statistical inferences about the entire population. Collecting a full dataset from the entire population can be resource-intensive and sometimes impractical, so researchers often rely on samples to generate insights.

Key Features:
  • Representativeness: A good sample should reflect the characteristics of the entire population. If it's not representative, any conclusions drawn may not be valid.
  • Size: The size of the sample affects the reliability of statistical calculations. Larger samples tend to provide more reliable estimates of population parameters.
  • Variability: Within the sample, variability is quantified using measures such as standard deviation, which represents how spread out the observations are in the sample.
Understanding sample data characteristics helps researchers draw better conclusions about the population from which the sample was drawn.
Standard Error
Standard error is a statistical term that measures the accuracy with which a sample distribution represents a population. It is specifically focused on the variability or spread of the sampling distribution of a statistic, most commonly the mean.

Essentials of Standard Error:
  • It quantifies how much the sample mean is likely to vary from the true population mean.
  • The formula for the standard error of the mean is given by \( SE = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the standard deviation of the population and \( n \) is the sample size.
  • Smaller standard errors suggest that the sample mean is a more accurate reflection of the actual population mean.
  • As the sample size increases, the standard error decreases, indicating more precise estimates with larger samples.
This concept is essential when making inferences about the population, especially when constructing confidence intervals.
Confidence Interval
A confidence interval is a range of values, derived from sample data, that is used to estimate an unknown population parameter. This range provides a degree of certainty or confidence on how close the sample statistic is to the actual parameter.

Elements of Confidence Interval:
  • A common choice for confidence level is 95%, meaning that we would expect the interval to contain the true parameter 95% of the time.
  • The formula for a confidence interval for the mean is \( \bar{x} \pm z \times SE \), where \( \bar{x} \) is the sample mean, \( z \) is the z-score corresponding to the chosen confidence level, and \( SE \) is the standard error.
  • Wider intervals may occur with smaller samples or higher desired levels of confidence, indicating less precision.
  • Interpreting a confidence interval involves acknowledging a balancing act between range (interval width) and certainty (confidence level).
Confidence intervals offer a clear method for understanding statistical findings' reliability and are widely used in hypothesis testing and estimations.

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Most popular questions from this chapter

Basketball shooting In college basketball, a shot made from beyond a designated arc radiating about 20 feet from the basket is worth three points, instead of the usual two points given for shots made inside that arc. Over his career, University of Florida basketball player Lee Humphrey made \(45 \%\) of his three-point attempts. In one game in his final season, he made only 3 of 12 three-point shots, leading a TV basketball analyst to announce that Humphrey was in a shooting slump. a. Assuming Humphrey has a \(45 \%\) chance of making any particular three-point shot, find the mean and standard deviation of the sampling distribution of the proportion of three-point shots he will make out of 12 shots. b. How many standard deviations from the mean is this game's result of making 3 of 12 three-point shots? c. If Humphrey was actually not in a slump but still had a \(45 \%\) chance of making any particular three-point shot, explain why it would not be especially surprising for him to make only 3 of 12 shots. Thus, this is not really evidence of a shooting slump.

Baseball hitting Suppose a baseball player has a 0.200 probability of getting a hit in each time at-bat. a. Describe the shape, mean, and standard deviation of the sampling distribution of the proportion of times the player gets a hit after 36 at- bats. b. Explain why it would not be surprising if the player has a 0.300 batting average after 36 at-bats.

Exit poll CNN conducted an exit poll of 1751 voters in the 2010 Senatorial election in New York between Charles Schumer and Jay Townsend. It is possible that all 1751 voters sampled happened to be Charles Schumer supporters. Investigate how surprising this would be, if actually \(65 \%\) of the population voted for Schumer, by a. Finding the probability that all 1751 people voted for Schumer. (Hint: Use the binomial distribution.) b. Finding the number of standard deviations that a sample proportion of 1.0 for 1751 voters falls from the population proportion of \(0.65 .\)

Other scenario for exit poll Refer to Examples 1 and 2 about the exit poll, for which the sample size was \(3889 . \mathrm{In}\) that election, \(40.9 \%\) voted for Whitman. a. Define a binary random variable \(X\) taking values 0 and 1 that represents the vote for a particular voter \((1=\) vote for Whitman and \(0=\) another candidate \()\) State its probability distribution, which is the same as the population distribution for \(X\). b. Find the mean and standard deviation of the sampling distribution of the proportion of the 3889 people in the sample who voted for Whitman.

Canada lottery In one lottery option in Canada (Source: Lottery Canada), you bet on a six-digit number between 000000 and \(999999 .\) For a \(\$ 1\) bet, you win \(\$ 100,000\) if you are correct. The mean and standard deviation of the probability distribution for the lottery winnings are \(\mu=0.10\) (that is, 10 cents) and \(\sigma=100.00\). Joe figures that if he plays enough times every day, eventually he will strike it rich, by the law of large numbers. Over the course of several years, he plays 1 million times. Let \(\bar{x}\) denote his average winnings. a. Find the mean and standard deviation of the sampling distribution of \(\bar{x}\). b. About how likely is it that Joe's average winnings exceed \(\$ 1,\) the amount he paid to play each time? Use the central limit theorem to find an approximate answer.
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