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Chapter 8: Problem 75

Multiple choice: Select the best answer for Exercises 75 to 78. One reason for using a t distribution instead of the standard Normal curve to find critical values when calculating a level C confidence interval for a population mean is that (a) z can be used only for large samples. (b) z requires that you know the population standard deviation S. (c) z requires that you can regard your data as an SRS from the population. (d) the standard Normal table doesn鈥檛 include confidence levels at the bottom. (e) a z critical value will lead to a wider interval than a t critical value.

Short Answer

Expert verified
Choice (b) is correct: z requires that you know the population standard deviation.

Step by step solution

01

Understand the purpose of t-distribution

The t-distribution is used in place of the standard normal distribution (z-distribution) when the population standard deviation is unknown and the sample size is small.
02

Analyze choice (a)

Choice (a) states that z can be used only for large samples. However, z can also be used for small samples if the population standard deviation is known. So, this choice is incorrect.
03

Analyze choice (b)

Choice (b) states that z requires that you know the population standard deviation. This is true because z-scores are used when the population standard deviation is known, which is why the t-distribution is used when it is unknown.
04

Analyze choice (c)

Choice (c) states that z requires you can regard your data as an SRS (simple random sample) from the population. This is a requirement for both z and t distributions, so this choice is not specific to justifying the use of t-distribution.
05

Analyze choice (d)

Choice (d) incorrectly states that the standard Normal table doesn鈥檛 include confidence levels at the bottom. This is not a valid reason for using the t-distribution over the z-distribution.
06

Analyze choice (e)

Choice (e) states that a z critical value will lead to a wider interval than a t critical value. This is incorrect; typically, t critical values lead to wider intervals due to accounting for extra uncertainty with smaller sample sizes.
07

Select the correct answer

Based on the analysis, choice (b) is the correct reason to use the t-distribution instead of the standard Normal curve when the population standard deviation is unknown.

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Key Concepts

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

Confidence Interval
A confidence interval is a range of values used to estimate the true population parameter. It provides a way of understanding what the possible values of the population parameter could be, considering the variability in a sample. Confidence intervals are built around a sample statistic, for instance, the sample mean. This interval gives you a range, and the confidence level tells you how likely it is that the true population parameter lies within this range.
  • The higher the confidence level, the wider the interval. This is because we are more certain that the population parameter lies within the interval.
  • If you're using a 95% confidence interval, it suggests that if you were to take 100 different samples and compute a confidence interval for each, approximately 95 of those intervals would include the population mean.
It is important to choose the appropriate distribution when calculating these intervals. This can depend on whether you know the population standard deviation and the sample size.
Population Mean
The population mean represents the average of all the values in a population. It's a parameter that indicates the central tendency of the data. In practice, it can be challenging to calculate the population mean directly because it's often impossible or impractical to collect data from every individual in a large population.
  • As a solution, we collect a sample and calculate the sample mean, which acts as an estimate of the population mean.
  • The formula for calculating a sample mean is the sum of all sample observations divided by the number of observations.
  • The accuracy of the sample mean in estimating the population mean improves with random sampling and larger sample sizes.
Remembering that the sample mean is just an estimate can help understand the variability and error involved in inferential statistics.
Z-Score
A z-score measures how many standard deviations an element is from the mean. It is used in statistics to determine the position of a specific data point within a dataset. When you're using z-scores, you're typically dealing with the standard normal distribution (also known as the z-distribution), which has a mean of 0 and a standard deviation of 1.
  • Z-scores are essential for calculating probabilities and creating confidence intervals when the population standard deviation is known.
  • However, when this parameter is unknown, we shift to using t-scores instead, as indicated in the original exercise and step-by-step solution.
  • To calculate a z-score, use the formula: \[ z = \frac{(X - \mu)}{\sigma} \] where \(X\) is the value in the data set, \(\mu\) is the mean of the population, and \(\sigma\) is the population standard deviation.
Using z-scores can help us understand outliers and the spread of our data quickly.
Population Standard Deviation
The population standard deviation is a measure of variability that indicates the average distance between each data point and the mean of the data set. In other words, it shows how much the data points differ from the average value. Knowing the population standard deviation is crucial for certain statistical calculations, such as when using z-scores.
  • When the population standard deviation is known, it helps in providing a more accurate assessment of variability and determining exact probabilities under the assumption of normality.
  • However, in many real-world scenarios, the population standard deviation is unknown. When this is the case, the t-distribution is used for statistical analysis.
  • The formula for calculating the population standard deviation (\(\sigma\)) is:\[\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (X_i - \mu)^2}\]where \(N\) is the number of observations, \(X_i\) are the data points, and \(\mu\) is the population mean.
Understanding the concept of standard deviation helps in appreciating the spread and distribution of data in statistics.

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Most popular questions from this chapter

NAEP scores Young people have a better chance of full-time employment and good wages if they are good with numbers. How strong are the quantitative skills of young Americans of working age? One source of data is the National Assessment of Educational Progress (NAEP) Young Adult Literacy Assessment Survey, which is based on a nationwide probability sample of households. The NAEP survey includes a short test of quantitative skills, covering mainly basic arithmetic and the ability to apply it to realistic problems. Scores on the test range from 0 to 500. For example, a person who scores 233 can add the amounts of two checks appearing on a bank deposit slip; someone scoring 325 can determine the price of a meal from a menu; a person scoring 375 can transform a price in cents per ounce into dollars per pound. \(^{4}\) Suppose that you give the NAEP test to an SRS of 840 people from a large population in which the scores have mean 280 and standard deviation S 60. The mean \(\overline{x}\) of the 840 scores will vary if you take repeated samples. (a) Describe the shape, center, and spread of the sampling distribution of \(\overline{x} .\) (b) Sketch the sampling distribution of \(\overline{x}\) . Mark its mean and the values one, two, and three standard deviations on either side of the mean. (c) According to the \(68-95-99.7\) rule, about 95\(\%\) of all values of \(\overline{x}\) lie within a distance \(m\) of the mean of the sampling distribution. What is \(m ?\) Shade the region on the axis of your sketch that is within \(m\) of the mean. (d) Whenever \(\overline{x}\) falls in the region you shaded, the population mean \(\mu\) lies in the confidence interval \(\overline{x} \pm m .\) For what percent of all possible samples does the interval capture \(\mu\) ?

Teens鈥� online profiles Over half of all American teens (ages 12 to 17 years) have an online profile, mainly on Facebook. A random sample of 487 teens with profiles found that 385 included photos of themselves.13 (a) Construct and interpret a 95% confidence interval for p. Follow the four- step process. (b) Is it plausible that the true proportion of American teens with profiles who have posted photos of themselves is 0.75? Use your result from part (a) to support your answer.

Multiple choice: Select the best answer for Exercises 75 to 78. A quality control inspector will measure the salt content (in milligrams) in a random sample of bags of potato chips from an hour of production. Which of the following would result in the smallest margin of error in estimating the mean salt content \(\mu ?\) (a) 90% confidence; n 25 (b) 90% confidence; n 50 (c) 95% confidence; n 25 (d) 95% confidence; n 50 (e) n 100 at any confidence level

Multiple choice: Select the best answer for Exercises 49 to 52. Most people can roll their tongues, but many can鈥檛. The ability to roll the tongue is genetically determined. Suppose we are interested in determining what proportion of students can roll their tongues. We test a simple random sample of 400 students and find that 317 can roll their tongues. The margin of error for a 95% confidence interval for the true proportion of tongue rollers among students is closest to (a) 0.008. (c) 0.03. (e) 0.208. (b) 0.02. (d) 0.04.

For Exercises 27 to 30, check whether each of the conditions is met for calculating a confidence interval for the population proportion p. AIDS and risk factors In the National AIDS Behavioral Surveys sample of 2673 adult heterosexuals, 0.2% had both received a blood transfusion and had a sexual partner from a group at high risk of AIDS. We want to estimate the proportion p in the population who share these two risk factors.
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