ShowerTime

= Shower Time Problem = Return to Central Limit Theorem

The notation for the normal distribution is N (population mean, standard deviation) The notation for the standard normal distribution is N(0,1)

Draw an SRS of size n from ANY population with mean mu and a finite standard deviation of sigma. When n is large, the sampling distribution of the sample mean is approximately Normal. As the sample size n increases without limit, the shape of the distribution of the sample means taken with replacement from a population with mean mu and standard deviation sigma will approach a normal distribution. This distribution will have a mean mu and standard deviation [sigma / sqrt(n) ].
 * Central Limit Theorem **
 * // CLT from a different source: //**


 * Americans spend on average 12.2 minutes in the shower. The standard deviation for the variable is 2.3 minutes and the variable is normally distributed. **

a. Find the probability that a randomly selected individual spends more than 14 minutes in the shower. z-score = (14-12.2)/ 2.3 = 0.78 P( x > 0.78) = 21.77% [Note in the first graph shown below, 20% of the individuals showered longer than 14 minutes.]

b. Find the probability that the mean time of a random sample of 35 Americans who shower is more than 14 minutes. z-score = (14-12.2) / [ 2.3/sqrt(35)] = 4.63 P( x-bar > 4.63) = 0

We had a class discussion about why the answer was zero when all of the other problems we completed in class were non-zero. Brent pointed that the standard deviation is small to begin with, so as it is reduced from 2.3 minutes to 0.39 minutes for the sample means, the z-score should be this high (4.63). The probability is essentially zero. Jacob suggested that we model this in TP, so I did.

How often will the mean shower time for a random sample of 35 Americans be greater than 14 minutes? I used the randomNormal(12.2, 2.3) formula with 35 cases. The first plot shows a graph of 35 individuals and the mean shower time of 11.784 minutes. Note in the first graph shown below, 20% of the individuals showered longer than 14 minutes.



This next plot shows another random sample of 35 American's shower times. Note that the mean is 12.08 and the SD is 2.6. These values are close to the population parameters. We see that 9 of the 35 people showered longer than 14 minutes.

From there, I captured the mean shower times for 1000 randomized groups of 35 people. As we look at the new plot, we notice that the range of the data is much smaller. The plots above (with individual people) have a range between around 6 minutes to almost 18 minutes. In the plot of the means of the groups of 35 individuals, the range is much smaller. In the simulation below, the minimum shower time is above 10.5 minutes and the maximum shower time is less than 14 minutes. This models the Central Limit Theorem. ** This distribution will have a mean mu and standard deviation [sigma / sqrt(n) ]. ** Note that the standard deviation for the distribution of the sample means shown with the SD Hat plot is 0.39 (for this set of random samples). In class, I refer to this as the (new improved SD). The proper name for this new standard deviation is the **standard error of the mean**.

The plot for this simulation shows that the number of sample means (x-bars) above 14 minutes is zero.

Another sample of 100 random samples of 35 individuals' shower times results in a SD of 0.41. The highest shower time for a group is close to 13 minutes.

I decided to try one more thing. I created a population of 1,000 people with a mean shower time of 12.2 minutes and a SD of 0.39. The plot shown below is similar to what we would expect with such a small SD. None of shower times were above 14 minutes.

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