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Chapter 7: Problem 5

Find the critical value \(z_{a / 2}\) that corresponds to the given confidence level. $$90 \%$$

Short Answer

Expert verified
1.645

Step by step solution

01

Determine the confidence level

The given confidence level is 90%, which means we have a 90% confidence interval.
02

Find the remaining percentage

The remaining percentage outside the confidence interval is ewline \(100\text{%}-90\text{%}=10\text{%}\).
03

Divide the remaining percentage by 2

Since the remaining 10% is split between the two tails of the distribution, divide it by 2:\(10\text{%}/2=5\text{%}\).ewline This means \(5\text{%}\) (or \(0.05\) in decimal form) lies in each tail.
04

Find the cumulative probability

The cumulative probability required to find the critical value at the tail is ewline \(1-0.05=0.95\).
05

Use the z-table

Look up the value in the standard normal (z) table for a cumulative probability of \(0.95\). The corresponding value in the z-table is approximately \(1.645\).

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Key Concepts

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

z-score
A z-score is a statistical measurement that describes a value's position relative to the mean of a group of values. It's measured in terms of standard deviations from the mean. For instance, a z-score of 1.645 means the value is 1.645 standard deviations above the mean.

Key Points:
  • A positive z-score indicates the value is above the mean.
  • A negative z-score indicates the value is below the mean.
  • Z-scores are used in hypothesis testing and confidence interval calculations.
They help in understanding how unusual or typical a certain value is within a distribution. In the context of confidence intervals, z-scores are used to establish critical values that determine the cut-off points for specific confidence levels, such as 90% or 95%.
confidence interval
A confidence interval is a range of values that's used to estimate the true value of a population parameter. It is associated with a certain probability (the confidence level), which measures the degree of certainty that the parameter lies within the interval.

Main Aspects:
  • The confidence level indicates the percentage of intervals, so if we have a 90% confidence level, 90 out of 100 such intervals would contain the true population parameter.
  • Critical values (z-scores) are used to determine the boundaries of the confidence interval.
  • The size of the confidence interval is influenced by the sample size and variability within the data.
In the solved exercise, a 90% confidence level is used, which corresponds to z-scores of 卤1.645, establishing the cut-off points for the 90% confidence interval.
standard normal distribution
The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. It's often represented with a bell-shaped curve.

Characteristics:
  • The total area under the curve equals 1, representing the total probability.
  • It's symmetric around the mean (0).
  • 68% of the data falls within 卤1 standard deviation from the mean.
  • 95% falls within 卤2 standard deviations, and nearly all (99.7%) falls within 卤3 standard deviations.
The standard normal distribution is critical in z-score calculations, as any normal distribution can be converted to a standard normal distribution using the z-score formula. This standardization makes it easier to compare different data sets and apply statistical methods.
cumulative probability
Cumulative probability refers to the probability that a variable takes a value less than or equal to a specified value. It represents the area under the probability distribution curve to the left of that value.

Key Details:
  • Cumulative probabilities range from 0 to 1.
  • They are essential for finding z-scores from the z-table, which provides cumulative probabilities for the standard normal distribution.
  • In the previous example, a cumulative probability of 0.95 means that 95% of the data lies below z = 1.645.
By understanding cumulative probabilities, students can determine critical values (like the z-score) needed for constructing confidence intervals. This concept is a cornerstone of many statistical methods and in estimating the likelihood of different outcomes.

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Most popular questions from this chapter

Finding Sample Size Instead of using Table \(7-2\) for determining the sample size required to estimate a population standard deviation \(\sigma,\) the following formula can be used $$n=\frac{1}{2}\left(\frac{z_{\alpha / 2}}{d}\right)^{2}$$ where \(z_{\alpha / 2}\) corresponds to the confidence level and \(d\) is the decimal form of the percentage error. For example, to be \(95 \%\) confident that \(s\) is within \(15 \%\) of the value of \(\sigma,\) use \(z_{\alpha / 2}=1.96\) and \(d=0.15\) to get a sample size of \(n=86 .\) Find the sample size required to estimate \(\sigma,\) assuming that we want \(98 \%\) confidence that \(s\) is within \(15 \%\) of \(\sigma .\)

Why does the bootstrap method require sampling with replacement? What would happen if we used the methods of this section but sampled without replacement?

Use the given sample data and confidence level. In each case, (a) find the best point estimate of the population proportion \(p ;(b)\) identify the value of the margin of error \(E ;(c)\) construct the confidence interval; (d) write a statement that correctly interprets the confidence interval. In a study of the accuracy of fast food drive-through orders, McDonald's had 33 orders that were not accurate among 362 orders observed (based on data from \(Q S R\) magazine). Construct a \(95 \%\) confidence interval for the proportion of orders that are not accurate.

Finding Confidence Intervals. In Exercises \(9-16,\) assume that each sample is a simple random sample obtained from a population with a normal distribution. Highway Speeds Listed below are speeds (mi/h) measured from southbound traffic on I-280 near Cupertino, California (based on data from SigAlert). This simple random sample was obtained at 3: 30 PM on a weekday. Use the sample data to construct a \(95 \%\) confidence interval estimate of the population standard deviation. Does the confidence interval describe the standard deviation for all times during the week? \(\begin{array}{rrrrrrrrr}57 & 61 & 54 & 59 & 58 & 59 & 69 & 60 & 67\end{array}\) \(62 \quad 61 \quad 61\)

Finding Confidence Intervals. In Exercises \(9-16,\) assume that each sample is a simple random sample obtained from a population with a normal distribution. a. The values listed below are waiting times (in minutes) of customers at the Jefferson Valley Bank, where customers enter a single waiting line that feeds three teller windows. Construct a \(95 \%\) confidence interval for the population standard deviation \(\sigma\) \(\begin{array}{rrrrr}7.1 & 7.3 & 7.4 & 7.7 & 7.7 & 7.7\end{array}\) \(6.5 \quad 6.6 \quad 6.7 \quad 6.8\)
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