Table of Contents
- 1 What is the probability that the sample mean is between 95 and 105?
- 2 What is the probability that an observation is at least 1 standard deviation below the mean?
- 3 How do you do the 68 95 and 99.7 rule?
- 4 What is the probability of the population mean μ lying between 36.6 2 mins and 36.6 2 mins?
- 5 What is the 95 rule in statistics?
- 6 What is the probability that the random variable will take one deviation?
- 7 Why can’t normal distribution be used to estimate binomials?
- 8 When is the distribution of sample means normal?
What is the probability that the sample mean is between 95 and 105?
68\%
Solution: The sample mean has expectation 100 and standard deviation 5. If it is approximately normal, then we can use the empirical rule to say that there is a 68\% of being between 95 and 105 (within one standard deviation of its expecation).
What is the probability that an observation is at least 1 standard deviation below the mean?
This rule tells us that around 68\% of the data will fall within one standard deviation of the mean; around 95\% will fall within two standard deviations of the mean; and 99.7\% will fall within three standard deviations of the mean.
How do you find the probability of an event in a normal distribution?
Follow these steps:
- Draw a picture of the normal distribution.
- Translate the problem into one of the following: p(X < a), p(X > b), or p(a < X < b).
- Standardize a (and/or b) to a z-score using the z-formula:
- Look up the z-score on the Z-table (see below) and find its corresponding probability.
How do you do the 68 95 and 99.7 rule?
Apply the empirical rule formula:
- 68\% of data falls within 1 standard deviation from the mean – that means between μ – σ and μ + σ .
- 95\% of data falls within 2 standard deviations from the mean – between μ – 2σ and μ + 2σ .
- 99.7\% of data falls within 3 standard deviations from the mean – between μ – 3σ and μ + 3σ .
What is the probability of the population mean μ lying between 36.6 2 mins and 36.6 2 mins?
95.4\%
Now, we can say that there is a 95.4\% probability that the Population Mean(μ) lies between (36.6–2, 36.6+2).
What is 3 standard deviation below the mean?
99.7\%
The Empirical Rule states that 99.7\% of data observed following a normal distribution lies within 3 standard deviations of the mean. Under this rule, 68\% of the data falls within one standard deviation, 95\% percent within two standard deviations, and 99.7\% within three standard deviations from the mean.
What is the 95 rule in statistics?
The 95\% Rule states that approximately 95\% of observations fall within two standard deviations of the mean on a normal distribution.
What is the probability that the random variable will take one deviation?
Using a table of values for the standard normal distribution, we find that P(–1 < Z ≤ 1) = 2 (0.8413) – 1 = 0.6826 Thus, there is a 0.6826 probability that the random variable will take on a value within one standard deviation of the mean in a random experiment.
What is the probability range of a normal distribution?
A table for the standard normal distribution typically contains probabilities for the range of values –∞ to x (or z)–that is, P(X ≤ x). This probability is the same as
Why can’t normal distribution be used to estimate binomials?
The discrepancy between the estimated probability using a normal distribution and the probability of the original binomial distribution is apparent. The criteria for using a normal distribution to estimate a binomial thus addresses this problem by requiring BOTH np AND n (1 − p) are greater than five.
When is the distribution of sample means normal?
To summarize, the distribution of sample means will be approximately normal as long as the sample size is large enough. This discovery is probably the single most important result presented in introductory statistics courses. It is stated formally as the Central Limit Theorem.