Problem 1 (20 pts). Normal distributions are used to approximate the sampling distribution of the sample proportion pˆ when sample size is large (i.e., when np ≥ 10, n(1 − p) ≥ 10). Similarly, we can use a normal distribution to approximate a Binomial distribution when sample size is large, as shown in the lecture notes. However, the accuracies of these approximations are unclear (i.e., how close are the approximate answers to the exact answers if exact answers are available)? Here, you are asked to evaluate the accuracies of the normal approximations in several situations, based on a random variable X which follows the Binomial distribution B(n,p).

You can easily obtain exact binomial probabilities from software R with command “pbinom” (some books also give exact binomial probability table. Calculating the exact binomial probabilities by hand is also feasible, though a bit tedious). For the following questions, you can obtain exact binomial probabilities using any of these approaches.

(a) (12 pts). Compute the following probabilities using both the Binomial distribution X ∼ B(n, p) (which gives an exact answer) and its normal approximation X ∼ N (np, np(1 − p)) (which gives an approximate answer), and compare the two answers to see if they are close or not:

(i) choose n = 10 and p = 0.2, and then compute P (X ≤ 3) using both methods; (ii) choose n = 10 and p = 0.4, and then compute P (X ≤ 3) using both methods; (iii) choose n = 50 and p = 0.2, and then compute P (X ≤ 8) using both methods; (iv) choose n = 50 and p = 0.4, and then compute P (X ≤ 8) using both methods;

Summarize the above results in a table and state your conclusions in no more than three sentences (i.e., in what cases the normal approximations seem most accurate – the exact answers and approx- imate answers are closest). (b) (8 pts). The sample proportion pˆ can be defined as pˆ = X/n, where X ∼ B(n, p). For the four cases (i)–(iv) in question (a), compute the probabilities P(pˆ ≤ 0.3) using normal approximations (i.e., pˆ ∼ N (p, p(1 − p)/n)). Intuitively (or based on what you have observed in (a)), which ap- proximation(s) do you think may be the most accurate one? (Note: the sample proportion pˆ has no exact distribution available, so we have to use normal approximations.)

(a) Exact and normal approximationsR-Output1) Exact> pbinom(3,10,0.2) [1] 0.8791261 > pbinom(3,10,0.4) [1] 0.3822806 > pbinom(8,50,0.2) [1] 0.3073316 > pbinom(8,50,0.4) [1].. Get **Nursing Homework help** with **HelpHub** .

** Why Choose HelpHub**

**Quality Researched Papers**

We always make sure that writers follow all your instructions precisely. You can choose your academic level: high school, college/university or professional, and we will assign a writer who has a respective degree.

**Qualified Writers**

We have hired a team of professional writers experienced in academic and business writing. Most of them are native speakers and PhD holders able to take care of any assignment you need help with.

**Unlimited Revisions**

If you think we missed something, send your order for a free revision. You have 10 days to submit the order for review after you have received the final document. You can do this yourself after logging into your personal account.

**On Time Delivery**

All papers are always delivered on time. In case we need more time to master your paper, we may contact you regarding the deadline extension. We will always strive to deliver on time.

**Original & Confidential**

We use several writing tools checks to ensure that all documents you receive are free from plagiarism. Our editors carefully review all quotations in the text.

Our support agents are available 24 hours a day 7 days a week and committed to providing you with the best customer experience. Get in touch whenever you need any assistance.

Try it now!

How it works?

Follow these simple steps to get your paper done

Place your order

Fill in the order form and provide all details of your assignment.

Proceed with the payment

Choose the payment system that suits you most.

Receive the final file

Once your paper is ready, we will email it to you.

HelpHub Writing Services

No need to work on essay at night. Sleep tight, we will cover your back. We offer all kinds of essay writing services.

Essays

No matter what kind of academic paper you need and how urgent you need it, you are welcome to choose your academic level and the type of your paper at an affordable price. We take care of all your paper needs and give a 24/7 customer care support system.

Admissions

Admission Essays

An admission essay is an essay or other written statement by a candidate, often a potential student enrolling in a college, university, or graduate school. You can be rest assurred that through our service we will write the best admission essay for you.

Editing

Editing Support

Our academic writers and editors make the necessary changes to your paper so that it is polished. We also format your document by correctly quoting the sources and creating reference lists in the formats APA, Harvard, MLA, Chicago / Turabian.

Revision

Revision Support

If you think your paper could be improved, you can request a review. In this case, your paper will be checked by the writer or assigned to an editor. You can use this option as many times as you see fit. This is free because we want you to be completely satisfied.