A marksman's chance of hitting a target with each of his shots is 60%. (Assume that the shots are independent of each other.) If he fires 30 shots, what is the probability of his hitting the target at least 20 time?
First, I found the mean by multiplying the trials by the probability of successes (.60 by 30) and received 18. Then I found the standard deviation (square root of 30x.60x.40), square root of npq and received the answer 2.68. Then I set up my equation as (19.5-18/2.68. 19.5 because I was told to not use 18, as x should be greater than or equal to 5 I believe.) I got the answer 0.56. Then I looked up my answer on the z score table and found .7123 but I am not sure if this whole process is correct.
http://davidmlane.com/hyperstat/z_table.html
Your process is almost correct, but there is a small mistake in the equation setup.
Let's go through the steps again:
1. Find the mean (μ) using the formula:
μ = n * p = 30 * 0.60 = 18
The mean represents the average number of successful shots.
2. Find the standard deviation (σ) using the formula:
σ = √(n * p * q) = √(30 * 0.60 * 0.40) ≈ 2.92
Standard deviation measures how much the data varies or deviates from the mean.
3. To find the probability of hitting the target at least 20 times, you need to use the cumulative probability (also known as the cumulative distribution function or CDF).
Since we are looking for at least 20 successes, we sum up the probabilities from 20 to 30, inclusive.
P(x ≥ 20) = 1 - P(x < 20)
= 1 - Σ P(x = i), for i = 0 to 19
Here, Σ represents the sum.
4. Calculate the cumulative probability using the z-score.
z = (x - μ) / σ
where x is the number of successes you want to find the probability for, μ is the mean, and σ is the standard deviation.
5. Look up the cumulative probability in the z-score table or use a calculator to find it.
Now, let's plug in the values and calculate:
P(x ≥ 20) = 1 - P(x < 20)
= 1 - Σ P(x = i), for i = 0 to 19
The calculation might be a bit tedious, but you can use tables, calculators, or software to make it easier. The final result should be the probability of hitting the target at least 20 times.
Your overall approach is correct, but there are a few errors in your calculations.
Step 1: Calculate the mean (μ)
To find the mean, you multiply the number of trials (30) by the probability of success (0.60):
μ = 30 * 0.60 = 18
Step 2: Calculate the standard deviation (σ)
Now, calculate the standard deviation using the formula:
σ = √(n * p * q)
where n is the number of trials, p is the probability of success, and q is the probability of failure (1 - p).
In this case, n = 30, p = 0.60, and q = 1 - p = 0.40:
σ = √(30 * 0.60 * 0.40) ≈ 3.09
Step 3: Calculate the z-score
To find the probability of hitting the target at least 20 times, you need to calculate the z-score for the lower value (19.5) using the formula:
z = (x - μ) / σ
In this case, x = 19.5, μ = 18, and σ ≈ 3.09:
z = (19.5 - 18) / 3.09 ≈ 0.50
Step 4: Look up the z-score
Using a standard normal distribution table or a calculator, find the area under the curve to the left of the z-score you calculated. This represents the probability of hitting the target at most 19.5 times. Subtract this probability from 1 to get the probability of hitting the target at least 20 times.
However, there was a mistake in your calculation. A z-score of 0.50 corresponds to an area of 0.6915 under the curve (not 0.7123 as you mentioned). Subtracting this from 1 gives you the probability of hitting the target at least 20 times:
1 - 0.6915 ≈ 0.3085
So the correct probability of hitting the target at least 20 times when firing 30 shots is approximately 0.3085 or 30.85%.