Consider a scenario where measurement data is modeled by a normal distribution with a mean of 100. In one experiment, the standard deviation is 15, while in a subsequent experiment it is 10. Answer the following:
(a) For the N(100, 15^2) distribution, calculate the probability that an observation falls between 85 and 115.
(b) For the same two distributions, determine the probability that an observation is below 90, and compare the results.
Question
Given a normally distributed variable with mean 100 and standard deviation 15, calculate the probability that an observation falls between 85 and 115.
Hint
Hint
Hint
Compute the z-scores for 85 and 115 using the formula \(z = \frac{x-\mu}{\sigma}\) and then use the standard normal distribution table or your GDC.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
Approximately 68.3%
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Question
Now consider a second experiment where the variable follows a normal distribution with the same mean (100) but a standard deviation of 10. Calculate the probability that an observation falls below 90 for both distributions, and explain how the differences in standard deviation affect these probabilities.
Hint
Hint
Hint
For each distribution, compute the z-score for 90 using \(z = \frac{90-\mu}{\sigma}\) and then compare the corresponding probabilities. Consider how a smaller standard deviation concentrates data closer to the mean.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
For N(100,15^2): Approximately 25.2%; For N(100,10^2): Approximately 15.9%. The distribution with the smaller standard deviation (10) has a lower probability of an observation falling below 90 because it is less spread out around the mean.
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Answer Explanation
Answer Explanation
