/*! 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 27 A certain chromosome defect occu... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 27

A certain chromosome defect occurs in only 1 out of 200 adult Caucasian males. A random sample of \(n=100\) adult Caucasian males is to be obtained. a. What is the mean value of the sample proportion \(p\), and what is the standard deviation of the sample proportion? b. Does \(p\) have approximately a normal distribution in this case? Explain. c. What is the smallest value of \(n\) for which the sampling distribution of \(p\) is approximately normal?

Short Answer

Expert verified
The obtained mean value of the sample proportion is 0.5 and its standard deviation is 0.0704. The sample proportion is said to have a normal distribution and the smallest value for n that admits an approximately normal distribution is n = 1005.

Step by step solution

01

Find the Mean and Standard Deviation

In a binomial distribution, which is the case here since the adult Caucasian males can either have the chromosome defect or not, the mean of the distribution is found by \(np\) and the standard deviation by \(\sqrt{np(1-p)}\).\nWhere \(p = \frac{1}{200} = 0.005\) is the probability of a male having the defect and \(n = 100\) is the sample size.\nThus, the mean and standard deviation of the sample proportion can be calculated as follows:\nMean: \(np = 100 * 0.005 = 0.5\)\nStandard Deviation: \(\sqrt{np(1-p)} = \sqrt{100 * 0.005 * 0.995} = 0.0704\)
02

Check if the Distribution is Normal

The normality of a binomial distribution depends on the success and failure cases. It is given that both \(np\) and \(n(1-p)\) should be greater or equal than 5. As calculated earlier:\nnp=0.5 and\nn(1-p)=99.5.\nSince both are higher than 5, the distribution can be considered as approximately normal.
03

Find minimal N for Normal Distribution

To satisfy normality for the smallest sample size, both \(np\) and \(n(1 - p) \geq 5\). \nSince \(p = 0.005\), solving the inequalities would yield: \nFor np\(\geq 5, n \geq \frac{5}{0.005} = 1000\),\nFor n(1-p)\(\geq 5, n \geq \frac{5}{1-0.005} = 1005\).\nThus, the smallest value of n for which the sampling distribution will be approximately normal is n = 1005.

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!

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

A random sample is selected from a population with mean \(\mu=100\) and standard deviation \(\sigma=10 .\) Determine the mean and standard deviation of the \(\bar{x}\) sampling distribution for each of the following sample sizes: a. \(n=9\) b. \(n=15\) c. \(n=36\) d. \(n=50\) c. \(n=100\) f. \(n=400\)

Suppose that a sample of size 100 is to be drawn from a population with standard deviation \(10 .\) a. What is the probability that the sample mean will be within 2 of the value of \(\mu\) ? b. For this example \((n=100, \sigma=10)\), complete each of the following statements by computing the appropriate value: i. Approximately 95% of the time, \(\bar{x}\) will be within _____ of \(\mu .\) ii. Approximately 0.3% of the time, \(\bar{x}\) will be farther than _____ from\(\mu .\)

An airplane with room for 100 passengers has a total baggage limit of 6000 lb. Suppose that the total weight of the baggage checked by an individual passenger is a random variable \(x\) with a mean value of \(50 \mathrm{lb}\) and a standard deviation of \(20 \mathrm{lb}\). If 100 passengers will board a flight, what is the approximate probability that the total weight of their baggage will exceed the limit? (Hint: With \(n=100\), the total weight exceeds the limit when the average weight \(\bar{x}\) exceeds \(6000 / 100\).)

A manufacturer of computer printers purchases plastic ink cartridges from a vendor. When a large shipment is received, a random sample of 200 cartridges is selected, and each cartridge is inspected. If the sample proportion of defective cartridges is more than \(.02\), the entire shipment is returned to the vendor. a. What is the approximate probability that a shipment will be returned if the true proportion of defective cartridges in the shipment is .05? b. What is the approximate probability that a shipment will not be returned if the true proportion of defective cartridges in the shipment is .10?

The nicotine content in a single cigarette of a particular brand has a distribution with mean \(0.8 \mathrm{mg}\) and standard deviation \(0.1 \mathrm{mg}\). If 100 of these cigarettes are analyzed, what is the probability that the resulting sample mean nicotine content will be less than \(0.79 ?\) less than \(0.77\) ?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

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.