/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 50 The delivery times for all food ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 50

The delivery times for all food orders at a fast-food restaurant during the lunch hour are normally distributed with a mean of \(7.7\) minutes and a standard deviation of \(2.1\) minutes. Find the probability that the mean delivery time for a random sample of 16 such orders at this restaurant is a. between 7 and 8 minutes b. within 1 minute of the population mean c. less than the population mean by 1 minute or more

Short Answer

Expert verified
a. The probability that the mean delivery time is between 7 and 8 minutes can be found by looking up the probabilities for the calculated z-scores in a z-table and subtracting the probabilities. b. The probability that the mean delivery time is within 1 minute of the population mean can be found similarly. c. The probability that the mean delivery time is less than the population mean by 1 minute or more can be found by looking up the corresponding z-score in the z-table.

Step by step solution

01

Calculate the Standard Error

The standard error (SE) is the standard deviation divided by the square root of the sample size \(n\). Here, the standard deviation \(\sigma\) is 2.1 and the sample size \(n\) is 16. This gives us a standard error of \(\sigma / \sqrt{n} = 2.1 / \sqrt{16} = 0.525\).
02

Calculate and Find the Probability for a

We want the probability that the mean delivery time is between 7 and 8 minutes. We calculate the z-scores for these values using the formula \(z = (X - \mu) / SE\), where \(X\) is the value of interest, \(\mu\) is the population mean, and \(SE\) is the standard error. Here, we'll use \(\mu = 7.7\) and \(SE = 0.525\). For \(X = 7\), the z-score is \(z_7 = (7 - 7.7) / 0.525 = -1.33\), and for \(X = 8\), the z-score is \(z_8 = (8 - 7.7) / 0.525 = 0.57\). Now we look up these z-scores in the z-table (or using a standard normal distribution calculator) to get the probabilities \(P(z < -1.33)\) and \(P(z < 0.57)\). The probability we're interested in is the difference between these, \(P(7 < x < 8) = P(z < 0.57) - P(z < -1.33)\).
03

Calculate and Find the Probability for b

We want the probability that the mean delivery time is within 1 minute of the population mean. This is equivalent to finding the probability that the delivery time is between 6.7 and 8.7 minutes. We can use the same method as in step 2, calculating z-scores for these values and then looking up the probabilities in the z-table. So, \(P(6.7 < x < 8.7) = P(z < 1.9) - P(z < -1.9)\).
04

Calculate and Find the Probability for c

We want the probability that the mean delivery time is less than the population mean by 1 minute or more. This means we want the probability that the delivery time is less than 6.7 minutes. We can find the z-score for this value as before, giving \(z_{6.7} = (6.7 - 7.7) / 0.525 = -1.9\). The probability we want is \(P(x < 6.7) = P(z < -1.9)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

normal distribution
The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, known as the Gaussian distribution. It is incredibly important in statistics because it describes how the values of a variable are distributed.
In a normal distribution:
  • The mean, median, and mode are all equal and located at the center of the distribution.
  • The curve is symmetric about the mean.
  • About 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.
For the fast-food restaurant problem, the delivery times are normally distributed with a mean (\(\mu\)) of 7.7 minutes and a standard deviation (\(\sigma\)) of 2.1 minutes. This means most delivery times cluster around the average, with fewer times as you move farther from the mean.
standard error
The standard error (SE) measures how much the sample mean (\(\bar{x}\)) is expected to vary from the true population mean (\(\mu\)). It's essentially the standard deviation of the sampling distribution of the mean.
It is calculated using the formula:\[SE = \frac{\sigma}{\sqrt{n}}\]where \(\sigma\) is the standard deviation of the population, and \(n\) is the sample size.For our example with the fast-food delivery times, we calculated the standard error to be 0.525 by dividing the population standard deviation of 2.1 minutes by the square root of the sample size, which is 16. The smaller the standard error, the more accurately the sample mean estimates the population mean.
z-scores
Z-scores, or standard scores, are used to describe the position of a value within a normal distribution. They help us understand how far a specific value is from the mean in terms of standard deviations.
You calculate a z-score with the formula:\[z = \frac{X - \mu}{SE}\]where \(X\) is the value of interest, \(\mu\) is the population mean, and \(SE\) is the standard error.In our exercise, we calculated z-scores to find the probability of different delivery time scenarios:
  • For a 7-minute delivery: \(z_7 = \frac{7 - 7.7}{0.525} = -1.33\)
  • For an 8-minute delivery: \(z_8 = \frac{8 - 7.7}{0.525} = 0.57\)
These z-scores help us reference a standard normal distribution table to find the probability associated with the delivery times.
population mean
The population mean (\(\mu\)) is a measure of central tendency that represents the average of all values in a population. It's a fundamental concept in probability and statistics, providing a summary of a data set.
In the context of our problem, the population mean of the delivery times is 7.7 minutes. This value helps us determine whether other values, such as sample means or individual delivery times, are typical or unusual.Understanding the population mean is essential when comparing an individual or sample result to the whole population. It acts as a benchmark for evaluating how "normal" a data point is with respect to the rest of the data.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The Toyota Prius hybrid car is estimated to get an average of 50 miles per gallon (mpg) of gas. However, the gas mileage varies from car to car due to a variety of conditions, driving styles, and other factors and has been reported to be as high as \(70 \mathrm{mpg}\). Suppose that the distribution of miles per gallon for Toyota Prius hybrid cars has a mean of \(50 \mathrm{mpg}\) and a standard deviation of \(5.9 \mathrm{mpg}\). Find the probability that the average miles per gallon for 38 randomly selected Prius hybrid cars is a. more than \(51.5\) b. between 48 and 51 c. less than 53 d. greater than the population mean by \(2.5\) or more

A survey of all medium- and large-sized corporations showed that \(64 \%\) of them offer retirement plans to their employees. Let \(\hat{p}\) be the proportion in a random sample of 50 such corporations that offer retirement plans to their employees. Find the probability that the value of \(\hat{p}\) will be a. between \(.54\) and 61 b. greater than . 71

A certain elevator has a maximum legal carrying capacity of 6000 pounds. Suppose that the population of all people who ride this elevator have a mean weight of 160 pounds with a standard deviation of 25 pounds. If 35 of these people board the elevator, what is the probability that their combined weight will exceed 6000 pounds? Assume that the 35 people constitute a random sample from the population.

Mong Corporation makes auto batteries. The company claims that \(80 \%\) of its LL70 batteries are good for 70 months or longer. Assume that this claim is true. Let \(\hat{p}\) be the proportion in a sample of 100 such batteries that are good for 70 months or longer. a. What is the probability that this sample proportion is within \(.05\) of the population proportion? b. What is the probability that this sample proportion is less than the population proportion by \(.06\) or more? c. What is the probability that this sample proportion is greater than the population proportion by 07 or more?

For a population, \(\mu=46\) and \(\sigma=10\). a. For a sample selected from this population, \(\mu_{\bar{x}}=46\) and \(\sigma_{\bar{x}}=2.0 .\) Find the sample size. Assume \(n / N \leq .05\). b. For a sample selected from this population, \(\mu_{\bar{x}}=46\) and \(\sigma_{\bar{x}}=1.6 .\) Find the sample size. Assume \(n / N \leq .05\).
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.