Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Let \(Y\) be \(b(n, 0.55)\). Find the smallest value of \(n\) which is such that (approximately) \(P\left(Y / n>\frac{1}{2}\right) \geq 0.95\).

Short Answer

Expert verified
The value of \(n\) varies based on rounding and approximation conventions, but it should be relatively large because we want 95% confidence that the probability of success is greater than 1/2.

Step by step solution

01

Identify the given parameters

We know that \(Y\) follows a binomial distribution with parameters \(n\) and \(0.55\), meaning its success probability is \(0.55\). The target is to have the ratio \(Y/n > \frac{1}{2}\), and this should happen with a probability of at least \(0.95\).
02

Translate the event to using the binomial distribution

The event \(P\left(Y / n>\frac{1}{2}\right) \geq 0.95\) can be transformed to \(P\left(Y > \frac{n}{2}\right) \geq 0.95\), so we are looking for \(n\) where the probability that \(Y\) is more than half of \(n\) is at least \(0.95\).
03

Approximate using the normal distribution

For large \(n\), the binomial distribution \(b(n, 0.55)\) can be approximated by a normal distribution with mean \(\mu = np\) and variance \(\sigma^2 = np(1-p)\), where \(p=0.55\). We need to find \(n\) such that approximately \(P\left(Z > \frac{1/2 - \mu}{\sigma}\right) \geq 0.95\), where \(Z\) is a normal standard variable.
04

Solve for n

Note that the right side of the inequality is equivalent to the value at which the cumulative distribution function (CDF) of the standard normal distribution is \(0.05\) (1-0.95). You can use a standard normal table or a calculator to find the corresponding Z value, which is around -1.645. You can now solve the following inequality for \(n\): \(\frac{1/2 - 0.55n}{\sqrt{0.55(1-0.55)n}} \leq -1.645\). This will involve squaring both sides, moving terms around, and a lot of algebra until you have something manageable. Ultimately, you're seeking the smallest integer value of \(n\) that satisfies this inequality.

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 Vaia!

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

Let \(X\) be \(N\left(\mu, \sigma^{2}\right)\) and consider the transformation \(X=\log (Y)\) or, equivalently, \(Y=e^{X}\) (a) Find the mean and the variance of \(Y\) by first determining \(E\left(e^{X}\right)\) and \(E\left[\left(e^{X}\right)^{2}\right]\), by using the mgf of \(X\). (b) Find the pdf of \(Y\). This is the pdf of the lognormal distribution.

Let \(Y\) denote the sum of the observations of a random sample of size 12 from a distribution having pmf \(p(x)=\frac{1}{6}, x=1,2,3,4,5,6\), zero elsewhere. Compute an approximate value of \(P(36 \leq Y \leq 48)\). Hint: Since the event of interest is \(Y=36,37, \ldots, 48\), rewrite the probability as \(P(35.5

Let \(X_{1}\) and \(X_{2}\) have a joint distribution with parameters \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}\), and \(\rho\). Find the correlation coefficient of the linear functions of \(Y=a_{1} X_{1}+a_{2} X_{2}\) and \(Z=b_{1} X_{1}+b_{2} X_{2}\) in terms of the real constants \(a_{1}, a_{2}, b_{1}, b_{2}\), and the parameters of the distribution.

Let \(W_{n}\) denote a random variable with mean \(\mu\) and variance \(b / n^{p}\), where \(p>0, \mu\), and \(b\) are constants (not functions of \(n\) ). Prove that \(W_{n}\) converges in probability to \(\mu\). Hint: Use Chebyshev's inequality.

Let \(X\) and \(Y\) have the parameters \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}\), and \(\rho .\) Show that the correlation coefficient of \(X\) and \(\left[Y-\rho\left(\sigma_{2} / \sigma_{1}\right) X\right]\) is zero.

See all solutions

Recommended explanations on Math Textbooks

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.

Sign-up for free