sample standard deviation = 3
sample mean = 68
n=30
what is the probability that one student selected at random from the sample will have a height within one sample standard deviation of the sample mean?
a. 1.00
b. 0.73
c. 0.57
d. 0.23
mean ± 1 SD = 68.26%
This is true for any normal distribution. Is there something you are not telling me?
There is a picture of a histogram. the numbers on the x-axis are: 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, and 76. the numbers on the t-axis are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. there are additional symbols: sigma x =2100, sigma x^2=142540, and Sx=3
To find the probability that a student selected at random from the sample will have a height within one sample standard deviation of the sample mean, we need to use the concept of the standard normal distribution.
1. Calculate the standard error: The standard error is equal to the sample standard deviation divided by the square root of the sample size. In this case, the sample standard deviation is 3 and the sample size is 30.
Standard Error = 3 / sqrt(30) ≈ 0.5477
2. Find the Z-scores: A Z-score measures the number of standard deviations a data point is from the mean. Since we want to find the probability within one standard deviation of the mean, we need to find the Z-scores for x = 68 - 1 * Standard Error and x = 68 + 1 * Standard Error.
Z1 = (68 - 1 * 0.5477 - 68) / 0.5477 ≈ -0.5477
Z2 = (68 + 1 * 0.5477 - 68) / 0.5477 ≈ 0.5477
3. Look up the probabilities: Using a standard normal distribution table or a calculator, we can find the probabilities associated with the Z-scores. The area between Z = -0.5477 and Z = 0.5477 represents the probability within one sample standard deviation of the sample mean.
Using a Z-table or calculator, we find that the probability associated with Z = -0.5477 is approximately 0.2939, and the probability associated with Z = 0.5477 is approximately 0.7071.
4. Calculate the final probability: To find the probability within one standard deviation, we subtract the probability associated with Z = -0.5477 from the probability associated with Z = 0.5477.
Final Probability = 0.7071 - 0.2939 ≈ 0.4132
Therefore, the probability that one student selected at random from the sample will have a height within one sample standard deviation of the sample mean is approximately 0.4132.
The closest option to this value is option "c. 0.57".