Statistical methods and applications

frequency distribution below shows the distribution for suspended solid concentration (in ppm) in river water of 50 different waters collected in September 2011.

Concentration (ppm)

Frequency

What percentage of the rivers had suspended solid concentration greater than or equal to 70?

Total samples (N) =50. (7+2+2)/50=0.22. 22% have a concentration of 70 or greater.

Calculate the mean of this frequency distribution.

Midpoint for each concentration group is determined and multiplied by frequency; results summed and divided by N (50). Mean = 57.1

In what class interval must the median lie? Explain your answer. (You don't have to find the median)

The median must lie in the 50-59 interval as this is where the middle data points (25 and 26) would fall in this data set of 50 points (there are 17 point before this group and 23 after; though on the lower end, this group contains the median data points).

Assume that the smallest observation in this dataset is 20. Suppose this observation were incorrectly recorded as 2 instead of 20. Will the mean increase, decrease, or remain the same? Will the median increase, decrease or remain the same? Explain.

The mean of the raw data set -- that is, not of the frequency distribution -- would change based on this incorrect record somewhat significantly. The mean of the frequency distribution would change very slightly if the one observation in the 20-29 category simply disappeared, and even more slightly if the lowest group was altered to include this point (the midpoint used to estimate the mean would drop to 14.5). The median would remain the same, however, as moving the lowest data point would not change the order of the data or the position/identity of the central data point(s).

Refer to the following information for Questions 5 and 6.

A coin is tossed 4 times. Let a be the event that the first toss is heads. Let B. be the event that the third toss is heads.

5. What is the probability that the third toss is heads, given that the first toss is heads?

If it is already given that the first toss is heads, there is a 0.5 probability that the third toss will be heads -- the same as for any standard coin toss. Though the overall probability of both a and B. occurring is 0.25 (0.5*0.5), knowing a has already occurred gives B. its natural and independent probability.

6. Are a and B. independent? Why or why not? Each coin toss is independent as it is not influenced by previous tosses -- a previous heads does not actually change the coin or the toss in such a way as to affect the outcome. The probability that both a and B. will occur is different from the probabilities that a will occur and that B. will occur.

Refer to the following data to answer questions 7 and 8. Just the answer

A random sample of song playing times in seconds is as follows:

7. Find the standard deviation. 28.65

8. Are any of these playing times considered unusual in the sense of our textbook? Explain. Does this differ with your intuition? Explain. The longest playing time, 293 seconds, could be considered somewhat unusual in that it is more than 1.5 standard deviations away from the mean, meaning more than 93% of data in the population (if normally distributed) would fall below this level -- 93% of songs would be shorter. Intuition says that this is not the case, and that though the song is longer this is simply part of the variation that exists in music.

Refer to the following situation for Questions 9, 10, and 11.

The boxplots below show the real estate values of single family homes in two neighboring cities, in thousands of dollars.

For each question, give your answer as one of the following: (a) Tinytown; (b) BigBurg; (c) Both cities have the same value requested; (d) it is impossible to tell using only the given information. Please explain answer in each case.

9. Which city has greater variability in real estate values? (b) BigBurg