Question - Understanding Normal Distribution Probabilities
mathothernormal distribution probabilitiesmean and standard deviationz-score formulastandard normal distribution table
Solution:
This image presents a series of questions related to a normal distribution with a mean (μ) of 50 and a standard deviation (σ) of 4. The problem refers to a sample (n) of 100. The tasks are to find probabilities for certain values of X, which seems to represent a random variable. Here's how each question is addressed:a. The probability that X is less than 49:From the image, it looks like the probability, `P(X < 49)`, is given as 0.0062.b. The probability that X is between 49 and 50.5:The probability, `P(49 < X < 50.5)`, appears to be 0.8882.c. The probability that X is above 50.6:The probability, `P(X > 50.6)`, is shown as 0.0688.d. The value above which there is a 40% chance that X falls:The value of X, such that `P(X > X_value) = 0.40`, is given as 50.1012.These probabilities might have been calculated using a standard normal distribution table (Z-table) or a software that computes these values based on the Z-score formula. The Z-score is calculated by taking the difference between a value and the mean, then dividing by the standard deviation. In the case of part d, this Z-score would correspond with the 60th percentile value when looking up the normal distribution (since having 40% above the value means 60% is below it), and then this Z-score would be converted back to the specific value using the mean and standard deviation of the distribution.
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