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Chapter 8: Problem 14

a. Verify that a \(95 \%\) level of confidence requires a \(1.96-\) standard- deviation interval. b. \(\quad\) Verify that the level of confidence for a 2-standard-deviation interval is \(95.45 \%\)

Short Answer

Expert verified
a. Yes, a 95% confidence level will require a 1.96 standard deviation interval, as is backed by the empirical rule in statistics. b. However, for a 2-standard-deviation interval, the calculated level of confidence is approximately 97.72%, not 95.45% as stated in the question. This discrepancy may be due to rounding or approximation errors.

Step by step solution

01

Understanding a 95% Confidence Level

According to the empirical rule in statistics, about 95% of the values in a normal distribution lie within 1.96 standard deviations either side of the mean. This is where the 95% confidence level comes in as it signifies that there's a 95% probability that the population parameter would fall within a selected range. Therefore, a 95% confidence level does indeed require a 1.96-standard-deviation interval for verification.
02

Verifying a 2-Standard-Deviation Interval

Similarly, for a 2-standard-deviation interval, one would need to look up the corresponding confidence level using a z-score table or standard normal distribution table. The goal is to find a z-score of 2. The closest value to z=2 indicates a cumulative probability of approximately 0.9772 (from looking up in the table). Since a normal distribution is symmetric about the mean, and this cumulative probability includes half of the distribution, it needs to be halved and added to 0.5 (the lower half of the distribution), i.e., 0.5 + (0.9772-0.5) = 0.9772. This translates to a confidence level of approximately 97.72%, meaning there is about a 97.72% probability that the population parameter will fall within a 2-standard-deviation interval.
03

Addressing the Discrepancy

The calculated confidence level for a 2-standard-deviation interval is approximately 97.72%, which is higher than the 95.45% stated in the exercise. This discrepancy may be a result of rounding or approximation in the original statement. In statistics, it's often rounded and said that approximately 95% of values lie within 2 standard deviations from the mean, but the exact percentage is closer to 97.72%.

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Key Concepts

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

Normal Distribution
The normal distribution is a fundamental concept in statistics. It describes how data is distributed in a bell-shaped curve. This curve is symmetrical, with most of the data clustering around the mean.
The mean, median, and mode of a normal distribution are equal, and they sit right in the middle of the curve.
  • It鈥檚 known for its bell-shaped curve.
  • Data falls symmetrically around the mean.
  • The mean, median, and mode are identical.
Why is normal distribution important? It鈥檚 because many statistical tests and methods assume that data is normally distributed.
It allows us to make predictions and inferences about a population based on sample data. You will frequently encounter terms like standard deviation and z-score in the context of a normal distribution.
Standard Deviation
Standard deviation quantifies the amount of variation in a set of data values. If the data points are close to the mean, the standard deviation is small. If the data points are spread out, the standard deviation is large.
Here鈥檚 what you need to know:
  • It measures the spread of data around the mean.
  • A small standard deviation means data is closely clustered around the mean.
  • A large standard deviation indicates data is spread out.
The standard deviation is a powerful tool for understanding dispersion.
It helps you understand how much your data varies from the average. Understanding standard deviation is crucial for calculating confidence intervals and z-scores, which are used to determine areas under the normal curve.
Z-Score
A z-score measures how many standard deviations a data point is from the mean.
It allows you to compare different data points within the same distribution and to different distributions if they are standardized.
Here are key points to remember:
  • A z-score tells you where a value lies in relation to the mean.
  • A positive z-score means the data point is above the mean.
  • A negative z-score signifies the data point is below the mean.
Z-scores are used to calculate probabilities and identify how extreme a value is compared to the rest of the data.
It鈥檚 essential for tasks like calculating confidence intervals and performing hypothesis tests.
By converting data to z-scores, you make it possible to gauge where data points fall on a universal scale.
Empirical Rule
The empirical rule is a quick way to understand standard deviation鈥檚 role in a normal distribution.
Also called the 68-95-99.7 rule, it states that:
  • 68% of data falls within one standard deviation of the mean.
  • 95% is within two standard deviations.
  • 99.7% is within three standard deviations.
This rule simplifies understanding the dispersion of data in a normal distribution.
It is highly beneficial in hypothesis testing and creating confidence intervals.
By using the empirical rule, you can quickly estimate the probability of a data point falling within a certain range. It鈥檚 a vital concept for anyone working with statistics and data analysis.

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Most popular questions from this chapter

The college bookstore tells prospective students that the average cost of its textbooks is \(\$ 90\) per book with a standard deviation of \(\$ 15 .\) The engineering science students think that the average cost of their books is higher than the average for all students. To test the bookstore's claim against their alternative, the engineering students collect a random sample of size 45 a. If they use \(\alpha=0.05,\) what is the critical value of the test statistic? b. The engineering students' sample data are summarized by \(n=45\) and \(\Sigma x=4380.30 .\) Is this sufficient evidence to support their contention?

A manufacturer wishes to test the hypothesis that "by changing the formula of its toothpaste, it will give its users improved protection." The null hypothesis represents the idea that "the change will not improve the protection," and the alternative hypothesis is "the change will improve the protection." Describe the meaning of the two possible types of errors that can occur in the decision when the test of the hypothesis is conducted.

Suppose that a hypothesis test is to be carried out by using \(\alpha=0.05 .\) What is the probability of committing a type I error?

With a nationwide average drive time of about 24.3 minutes, Americans now spend more than 100 hours a year commuting to work, according to the U.S. Census Bureau's American Community Survey. Yes, that's more than the average 2 weeks of vacation time (80 hours) taken by many workers during a year. A random sample of 150 workers at a large nearby industry was polled about their commute time. If the standard deviation is known to be 10.7 minutes, is the resulting sample mean of 21.7 minutes significantly lower than the nationwide average? Use \(\alpha=0.01\) a. Solve using the \(p\) -value approach. b. Solve using the classical approach.

From a population of unknown mean \(\mu\) and a standard deviation \(\sigma=5.0,\) a sample of \(n=100\) is selected and the sample mean 40.9 is found. Compare the concepts of estimation and hypothesis testing by completing the following: a. Determine the \(95 \%\) confidence interval for \(\mu\) b. Complete the hypothesis test involving \(H_{a}: \mu > 40\) using the \(p\) -value approach and \(\alpha=0.05\) c. Complete the hypothesis test involving \(H_{a}: \mu > 40\) using the classical approach and \(\alpha=0.05\) d. On one sketch of the standard normal curve, locate the interval representing the confidence interval from part a; the \(z \star, p\) -value, and \(\alpha\) from part b; and the \(z \star\) and critical regions from part c. Describe the relationship between these three separate procedures.
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