Math 116 Chapter 11 Distribution of Sample Means Name______
Math 116 – Chapter 11 – Distribution of Sample Means Name______
1) Assume that cans of Coke are filled so that the actual amounts have a mean of 12.00 oz and a standard deviation of 0.11 oz. Assume the volumes of Coke cans is normally distributed.
a) What is the variable? Specify the characteristics of the distribution: shape, mean, standard deviation.
b) Find the theoretical probability that a can selected at random will have a VOLUME of at least 12.1 oz. (this is the process learned in chapter 3)
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c) Let’s simulate the experiment of selecting a can of a normal distribution with mean 12.0 oz and standard deviation 0.11 oz. Use randNorm(mu, sigma). (This is in the MATH, PRB menu). Do this ten times and record the obtained value each time (round to 3 decimal places). How many times did you obtain a number 12.1 or more? Use this result to find the experimental probability of selecting a can with at least 12.1 oz.
d) This will be done in class: Let’s collect class’ results to find the experimental probability of selecting at random one can that contains at least 12.1 oz.
Show organized and neat work below
2)) Assume that cans of Coke are filled so that the actual amounts have a mean of 12.00 oz and a standard deviation of 0.11 oz. Assume the volumes of Coke cans is normally distributed.
We select a sample of 36 Coke cans and the mean volume of the 36 cans is 12.1 oz.
a) Give the shape, mean and standard deviation of the distribution of sample means for samples of size 36.
b) Find the theoretical probability that a sample of 36 cans will have a mean amount of at least 12.1 oz.
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c) Simulate the experiment of selecting a sample of size 36 from a population with a mean of 12 oz and a standard deviation of 0.11 oz. and finding the mean of the sample. Do this five times.
Do: RandNorm(mu, sigma,36) →L1:mean(L1). OR mean(randNorm(mu, sigma,36)
How many times did you obtain a sample mean of at least 12.1 oz?
d) Let’s collect class’ results to find the experimental probability that a sample of 36 cans will have a mean amount of at least 12.1 oz. Use this result to find the experimental probability of selecting a can with at least 12.1 oz.
e) Interpret the results from part (b).
If the mean of the volumes of regular coke cans is 12 oz, in ______out of ______samples of size 36 we may observe a mean of at least 12.1 oz.
Based on this probability we can say that this is a ______event. (Complete with one of the following choices)
Very likelylikelyunlikelyvery unlikely
e) An actual sample of 36 Coke cans was selected and the observed sample mean was 12.1 oz. Is it reasonable to believe that the cans are actually filled with a mean of 12.00 oz?
f) If the mean is not 12.00 oz. as displayed in the Coke cans, are consumers being cheated? Explain.
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Givenx ~ ? (µ = 12 oz, σ = 0.11 oz)
a) The characteristics of the x-bar distribution for samples of size 36 are:
b) Find P(x-bar > 12.1)
Now find the probability: P( > 12.1) = P (z 5.49) = 1 – Area to the left ~ 1 – (close to 1) = close to 0
Normalcdf(12.1, 10^9, 12, 0.1833333333) = 2.5 * 10^-8 = 0.00000002
VERY SMALL PROBABILITY indicates a VERY UNLIKELY EVENT has happened!!!
If the population mean is 12 oz, it’s VERY UNUSUAL to observe a sample of 36 cans with a mean volume of 12.1 ounces or more.
Either the population mean is 12.0 ounces and a VERY RARE event happened, OR, the population from which the sample was selected has actually a mean higher than 12 oz, where a sample mean of 12.1 oz will be a more likely event.
On a COKE can we see advertised that the mean volume of the can is 12 oz. Are we sure that they are putting on average 12 oz. of coke in a can? NO.
How can we find out if what they say is true? We can’t go and measure the volumes of ALL cans produced, so we select a large sample from the cans and obtain the mean volume for the selected sample. THEN, we analyze how well the observed sample mean fits in the x-bar distribution for samples of size n.
IN OUR CASE, a sample of 36 cans was obtained and the x-bar observed was 12.1 oz.
HOW WELL DOES this x-bar fit in our x-bar distribution?
THIS x-bar is 5.49 standard errors to the right of the mean of 12. IT IS VERY FAR AWAY to the right. IT IS VERY UNUSUAL TO OBSERVE SUCH AN EVENT WHEN THE MEAN IS 12
Either something very unusual happened or the result suggests that the coke cans have on average MORE THAN 12 oz.