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Designed by Bill Bengen, the 4 percent rule says that a retiree may withdraw \(4 \%\) of savings during the first year of retirement, and then each year after that withdraw the same amount plus an adjustment for inflation. Under this rule, your retirement savings should be expected to last 30 years, which is longer than most retirements. (a) If your retirement savings is \(\$ 750,000\), how much may you withdraw in your first year of retirement if you want the retirement savings to last 30 years?(b) According to the American College of Financial Services, the proportion of people 60 to 75 years of age who believe it would be safe to withdraw 6 to 8 percent of their retirement savings annually is 0.16. Suppose you conduct a survey of twenty 60 to 75 year olds and ask them if it is safe to withdraw 6 to 8 percent of retirement savings annually if they wish their retirement savings to last 30 years. Explain why this is a binomial experiment. What are the values of \(n\) and \(p ?\) (c) In a random sample of twenty 60 to 75 year olds, what is the probability exactly 8 individuals will believe it is safe to withdraw 6 to 8 percent of retirement savings annually if they wish their retirement savings to last 30 years. (d) In a random sample of twenty 60 to 75 year olds, what is the probability fewer than 8 individuals will believe it is safe to (b) According to the American College of Financial Services, the proportion of people 60 to 75 years of age who believe it would be safe to withdraw 6 to 8 percent of their retirement savings annually is \(0.16 .\) Suppose you conduct a survey of twenty 60 to 75 year olds and ask them if it is safe to withdraw 6 to 8 percent of retirement savings annually if they wish their retirement savings to last 30 years Explain why this is a binomial experiment. What are the values of \(n\) and \(p ?\) (c) In a random sample of twenty 60 to 75 year olds, what is the probability exactly 8 individuals will believe it is safe to withdraw 6 to 8 percent of retirement savings annually if they wish their retirement savings to last 30 years. (d) In a random sample of twenty 60 to 75 year olds, what is the probability fewer than 8 individuals will believe it is safe to withdraw 6 to 8 percent of retirement savings annually if they wish their retirement savings to last 30 years. (e) Suppose you obtain a random sample of five hundred 60 to 75 year olds. Explain why the normal model may be used to describe the sampling distribution of \(\hat{p}\) the sample proportion of 60 to 75 year olds who believe it is safe to withdraw 6 to 8 percent of their retirement savings annually. Describe this sampling distribution. That is, find the shape, center, and spread of the sampling distribution of the sample proportion. (f) Use the normal model from part (e) to approximate the probability of obtaining a random sample of at least one hundred 60 to 75 years olds who believe it would be safe to withdraw 6 to 8 percent of their retirement savings annually assuming the true proportion is \(0.16 .\) Is this result unusual? Explain.

Step by step solution

01

Calculate the First Year Withdrawal

To determine the amount you may withdraw in the first year, use the 4 percent rule: \[ \text{First year withdrawal} = 0.04 \times 750000 \text{First year withdrawal} = 30000 \]

02

Identify the Characteristics of a Binomial Experiment

A binomial experiment must satisfy four conditions: 1. Fixed number of trials. 2. Each trial is independent. 3. There are only two outcomes (success or failure). 4. The probability of success is the same for each trial.For this problem, the fixed number of trials () is the 20 individuals surveyed, and the probability of success () is 0.16.

03

Define Parameters for Binomial Distribution

\( n = 20 \) and \( p = 0.16 \)

04

Calculate the Probability of Exactly 8 Believing 6-8% Withdrawal is Safe

Using the binomial probability formula: \[ P(X = 8) = {20 \choose 8} (0.16)^8 (0.84)^{12} \] Calculate the combination and probabilities to find the result.

05

Calculate the Probability of Fewer than 8 Believing 6-8% Withdrawal is Safe

Calculate the cumulative probability for values from 0 to 7 using the binomial probability formula: \[ P(X < 8) = \sum_{i=0}^{7} {20 \choose i} (0.16)^i (0.84)^{20-i} \]

06

Describe the Sampling Distribution for Sample Size 500

Shape: Approximately normal because \( np = 80 \) and \( n(1-p) = 420 \). Center: \( \mu_{\hat{p}} = p = 0.16 \) Spread: \( \sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.16 \times 0.84}{500}} \)

07

Calculate the Probability of at Least 100 Believing 6-8% Withdrawal is Safe

First, find the z-score for 100 out of 500: \[ z = \frac{\hat{p} - p}{\sigma_{\hat{p}}} = \frac{\frac{100}{500} - 0.16}{0.0168} = \frac{0.2 - 0.16}{0.0168} \] Using the z-score, use standard normal distribution tables or software to find the probability corresponding to z. Check if this probability is unusual (typically, if p < 0.05, it is considered unusual).

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Key Concepts

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

Binomial Distribution

The binomial distribution is used to model the number of successes in a fixed number of trials of a binary experiment. In a binomial experiment, there are only two possible outcomes: success (e.g., someone believes it is safe to withdraw 6-8% of retirement savings) and failure (e.g., someone does not believe this). To qualify as a binomial experiment, the following conditions must be met:

• Fixed number of trials (e.g., surveying 20 individuals).
• Each trial must be independent of the others.
• Only two outcomes per trial.
• The probability of success/failure remains constant for each trial (e.g., a 16% chance someone will agree).

In our exercise, we have 20 trials (surveying 20 individuals), which makes it a binomial experiment. The probability of someone agreeing is 0.16, and not agreeing is 0.84. We can calculate binomial probabilities using the formula: ..........