A fair coin is to be flipped 500 times. Use the “two-standard-deviation-rule” to calculate two numbers e and f such that the probability that the proportion of times that a head turns up will lie between e and f is approximately 95%.
I am totally confused and don't know where to start.
To calculate the values of e and f using the "two-standard-deviation rule" for this problem, we need to follow a few steps:
Step 1: Determine the sample size (n).
In this case, the sample size is given as 500 flips of the coin.
Step 2: Calculate the mean and standard deviation of the sample.
Since we're flipping a fair coin, the probability of getting a head is 0.5, and the probability of getting a tail is also 0.5. Therefore, the mean (µ) of the sample can be calculated as:
µ = n * p = 500 * 0.5 = 250
The standard deviation (σ) of a binomial distribution can be calculated using the formula:
σ = √(n * p * (1 - p))
σ = √(500 * 0.5 * 0.5) = √125 = 11.18 (approximated)
Step 3: Calculate the range for 2 standard deviations.
According to the "two-standard-deviation rule," approximately 95% of the values should be within 2 standard deviations of the mean.
To calculate the range, we multiply the standard deviation by 2 and add/subtract this value to/from the mean:
Range = 2 * σ = 2 * 11.18 = 22.36 (approximated)
Step 4: Calculate the values of e and f.
To determine the values of e and f, we subtract and add the range to/from the mean, respectively:
e = µ - Range = 250 - 22.36 = 227.64
f = µ + Range = 250 + 22.36 = 272.36
Therefore, the probability that the proportion of times a head turns up will lie between e and f is approximately 95%.
Note: The values of e and f represent proportions (in this case), so they should be expressed as percentages.
No worries, I can help you understand how to solve this problem step by step!
First, let's break down the problem. We want to find two numbers, e and f, such that the proportion of times a head turns up when flipping a fair coin 500 times will lie between e and f with approximately 95% probability.
To solve this, we need to use the concept of the standard deviation. In statistics, the standard deviation measures the amount of variation or dispersion in a set of numbers. It tells us how much the values differ from the mean (average).
Here's how we can approach the problem:
Step 1: Find the mean
In this case, since we have a fair coin, the probability of getting a head is 0.5, and the probability of getting a tail is also 0.5. Therefore, the expected value (mean) of the proportion of heads is:
mean = probability of head = 0.5
Step 2: Find the standard deviation
The standard deviation for a binomial distribution (flipping a coin) with probability p of success (getting a head) and n trials is given by:
standard deviation = sqrt(p * (1 - p) / n)
In this case, p = 0.5 (probability of getting a head) and n = 500 (number of coin flips). So, the standard deviation is:
standard deviation = sqrt(0.5 * (1 - 0.5) / 500)
Step 3: Calculate two standard deviations
Now, we want to find the range between e and f using the two-standard-deviation rule, which states that approximately 95% of the data falls within two standard deviations of the mean in a normal distribution.
We need to calculate:
e = mean - 2 * standard deviation
f = mean + 2 * standard deviation
Substituting the values we obtained in Step 1 and Step 2:
e = 0.5 - 2 * sqrt(0.5 * (1 - 0.5) / 500)
f = 0.5 + 2 * sqrt(0.5 * (1 - 0.5) / 500)
Now you can calculate e and f using a calculator, and these values will represent the range within which the proportion of heads are expected to fall with approximately 95% probability when flipping the fair coin 500 times.
I hope this explanation helps you understand how to approach this problem! Let me know if you have any further questions.