Question

below are the monthly rents paid by 30 students who live off campus.
620 620 620 820 590 480
580 920 630 510 610 565
450 630 540 550 740 810
490 510 650 580 600 390
620 690 710 730 610 645
%media:excel_png.ext% click here for the excel data file
(a) using excel, find the mean, median, mode, and standard deviation.
note: round your answers to 2 decimal places.
descriptive
statistics
data
mean
median
mode
standard deviation
(b) which measure or measures of central tendency are most appropriate for this data set?
the mean or the median because these measures are close in value, which suggests the data set is fairly
symmetric.
the median because the data set is strongly skewed.
the mode because the data is integer valued with a very small range.
(c) do the measures of central tendency agree?
no
yes
(e) use excel or megastat to sort and standardize the data. what is the z - score for the following rent value?
note: round your answer to 2 decimal places.
rent value z - score
810
(f) are there unusual data values?
no
yes
(g) using the empirical rule, do you think the data could be from a normal population?

Explanation:

Step1: Calculate the mean

To find the mean, sum all the data points and divide by the number of data points.
Sum = \(620+620+620+820+590+480+580+920+630+510+610+565+450+630+540+550+740+810+490+510+650+580+600+390+620+690+710+730+610+645 = 18265\)
Number of data points \(n = 30\)
Mean \(\bar{x}=\frac{18265}{30}\approx608.83\)

Step2: Calculate the median

First, sort the data in ascending order:
\(390,450,480,490,510,510,540,550,565,580,580,590,600,610,610,620,620,620,630,630,645,650,690,710,730,740,810,820,920\)
Since \(n = 30\) (even), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th values.
\(\frac{n}{2}=15\), \(\frac{n}{2}+1 = 16\)
Median \(=\frac{610 + 620}{2}=615.00\)

Step3: Calculate the mode

The mode is the value that appears most frequently. Here, \(620\) appears \(3\) times. So, mode \(=620.00\)

Step4: Calculate the standard deviation

Using the formula \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\)
First, calculate \((x_{i}-\bar{x})^{2}\) for each \(x_{i}\)
\(\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}=(620 - 608.83)^{2}+(620 - 608.83)^{2}+(620 - 608.83)^{2}+(820 - 608.83)^{2}+(590 - 608.83)^{2}+(480 - 608.83)^{2}+(580 - 608.83)^{2}+(920 - 608.83)^{2}+(630 - 608.83)^{2}+(510 - 608.83)^{2}+(610 - 608.83)^{2}+(565 - 608.83)^{2}+(450 - 608.83)^{2}+(630 - 608.83)^{2}+(540 - 608.83)^{2}+(550 - 608.83)^{2}+(740 - 608.83)^{2}+(810 - 608.83)^{2}+(490 - 608.83)^{2}+(510 - 608.83)^{2}+(650 - 608.83)^{2}+(580 - 608.83)^{2}+(600 - 608.83)^{2}+(390 - 608.83)^{2}+(620 - 608.83)^{2}+(690 - 608.83)^{2}+(710 - 608.83)^{2}+(730 - 608.83)^{2}+(610 - 608.83)^{2}+(645 - 608.83)^{2}\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=1039111.67\)
\(s=\sqrt{\frac{1039111.67}{29}}\approx189.60\)

Step5: Calculate the z - score for \(x = 810\)

The z - score formula is \(z=\frac{x-\bar{x}}{s}\)
\(z=\frac{810 - 608.83}{189.60}=\frac{201.17}{189.60}\approx1.06\)

Answer:

Descriptive Statistics	Data
Median	\(615.00\)
Mode	\(620.00\)
Standard Deviation	\(189.60\)
z - score for \(810\)	\(1.06\)

(b) The mean or the median because these measures are close in value, which suggests the data set is fairly symmetric.

(c) No

(f) Yes (since \(920\) has a relatively large deviation from the mean)

(g) No (due to the presence of outliers like \(920\) and the differences in central tendency measures)