I need help with this problem. I think I have some of the answers right but I'm not sure.

There is a data set consisting of 33 whole-number observations. Its five number summary is (16, 20, 22, 30, 46)

1. How many observations are stricty less than 22?

I think it is 16 because the median is 33 so if you think of the numbers on the "less than median" side ther should be 16 right?

2. How many observations are strictly less than 20?

How would I figure this out? 20 is a quartile so does that mean it is 33 divided by 4?

3. Is it possible that there is no observation equal to 20?

Well its a quartile so does it have to be part of the data? How do I figure this out?

4. Approximately where is the mean?

How dop I figure this out is I don't have the data? I only have the minimum, max, quartiles, and median.

Thank you!

*-------1-------M-------3-------*

With 33 observations, the median and quartiles fall exactly on real observations, as shown in the above figure.

1. The number could vary from 9 to 16, since observations 10 to 16 could be all 22's, and consequently are not strictly less than 22.

2. Similar response to question 1.

3. See figure above. It is impossible that there is no observation of 20, since the quartiles and the median are all actual observations (#9,#17,#25)

4. There are probably better estimates of the mean given quartiles, median and range.

Here's one[see ref.] that relies solely on the numbe of observations, n, the median m, and the extreme valus a,b.
μ = (a+2m+b)/4 + (a-2m+b)/4n
The second term is usually negligible for large n.
The estimate of the mean according to this formula is therefore
mean = (16+2*22+46)/4 + (16-2*22+46)/(4*33)
= 26.6

Ref:
http://www.biomedcentral.com/1471-2288/5/13

1. To determine the number of observations strictly less than 22, you need to consider the first quartile (Q1) of the data set, which is 20. Since Q1 represents the lower 25% of the data, the number of observations strictly less than 22 can be determined by subtracting Q1 from 33 and dividing it by 4.

Number of observations strictly less than 22 = (33 - Q1) / 4 = (33 - 20) / 4 = 13 / 4 = 3.25.

Since the number of observations must be a whole number, we round down to the nearest whole number. Therefore, the number of observations strictly less than 22 is 3.

2. To find the number of observations strictly less than 20, you need to consider the minimum value of the data set, which is 16. Since the data set only has whole-number observations, you can conclude that the number of observations strictly less than 20 is 16.

3. Yes, it is possible that there is no observation equal to 20. The five-number summary does not include every distinct value in the data set but provides information about specific percentiles (minimum, maximum, quartiles, and median). Therefore, there may be other values in the data set that are not included in the five-number summary, including the value 20.

4. It is not possible to determine the exact value of the mean with only the minimum, maximum, quartiles, and median given. However, since the five-number summary provides information about the spread of the data set, you can infer that the mean is approximately located near the median. In this case, the median is given as 30. So, you can estimate that the mean is close to 30.

1. To determine the number of observations strictly less than 22, you are correct that you can use the median as a reference point. Since the median is the 3rd quartile (Q3) and it falls at 22, you can assume that the upper half of the data set contains an equal number of observations. Therefore, the lower half of the data set also contains 16 observations that are strictly less than 22.

2. To determine the number of observations strictly less than 20, you can use the same logic as in the previous question. Since 20 is between the 1st quartile (Q1) and the median, you can assume that there are an equal number of observations below and above 20. Therefore, the number of observations strictly less than 20 would be half of the observations below the median, which is 16/2 = 8.

3. It is possible to not have any observation equal to 20. The quartiles (Q1, median, Q3) represent the values that divide the data into four equal parts. It does not necessarily mean that the data must have an observation at those exact values. So, it is possible that there are no observations exactly equal to 20.

4. Unfortunately, you cannot determine the exact location of the mean with just the minimum, maximum, quartiles, and median. However, based on the given information, you can estimate that the mean is close to the median since the five-number summary is relatively symmetric around the median. Keep in mind that this is just an estimation and may not be accurate without the actual data.