This problem depends on the following set up. Suppose you have a closed container (heated to a fixed temperature) full of a specific gas. We fix a specific gas particle within the container and measure its velocity as x m/s. The random variable x then follows a normal distribution with mean µ and σ = 5 m/s. a) Find z so that 93.8% of the area under the standard normal curve lies between −z and z. Note: If you choose not to do this part of the problem, use a confindence level of 95% in parts d, e, and f. b) Suppose that µ is 32.2 m/s. Find the percentage of time that the particle’s velocity is more that 35 m/s. c) Again, suppose that µ is 32.2 m/s. Find the percentage of time that the average velocity of the particle is more that 35 m/s if you take 9 random measurements. d) Suppose now that µ is unknown. A random sample of 9 measurements gives x¯ = 35 m/s. Calculate a 93.8% confidence interval. How likely is it that µ lies outside this interval? e) A random sample of 9 measurements gives ¯x = 35 m/s. Perform a two-tailed hypothesis test (with confidence level 93.8% ) to test the claim that µ = 32.2 m/s. f) Suppose that both µ and σ are unknown now. A random sample of 9 measurements gives ¯x = 35 m/s and s = 1.67 m/s. Perform a two-tailed hypothesis test (with confidence level 93.8% ) to test the claim that µ = 32.2 m/s.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
This problem depends on the following set up. Suppose
you have a closed container (heated to a fixed temperature) full of a specific gas.
We fix a specific gas particle within the container and measure its velocity as x m/s.
The random variable x then follows a
5 m/s.
a) Find z so that 93.8% of the area under the standard normal curve lies between
−z and z. Note: If you choose not to do this part of the problem, use a confindence
level of 95% in parts d, e, and f.
b) Suppose that µ is 32.2 m/s. Find the percentage of time that the particle’s
velocity is more that 35 m/s.
c) Again, suppose that µ is 32.2 m/s. Find the percentage of time that the average velocity of the particle is more that 35 m/s if you take 9 random measurements.
d) Suppose now that µ is unknown. A random sample of 9 measurements gives
x¯ = 35 m/s. Calculate a 93.8% confidence interval. How likely is it that µ lies
outside this interval?
e) A random sample of 9 measurements gives ¯x = 35 m/s. Perform a two-tailed
hypothesis test (with confidence level 93.8% ) to test the claim that µ = 32.2 m/s.
f) Suppose that both µ and σ are unknown now. A random sample of 9 measurements gives ¯x = 35 m/s and s = 1.67 m/s. Perform a two-tailed hypothesis
test (with confidence level 93.8% ) to test the claim that µ = 32.2 m/s.
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