Question

1. The following are the ages of a group of tourists visiting The Library of Congress:

36, 38, 33, 44, 37, 37, 38, 34, 34, 36, 34, 40, 42, 41, 35, 48, 31, 33, 46, 47, 33, 36, 37, 37, 34, 39, 35, 37, 36, 39, 39

1. Construct the Histogram of the ages with a class width of 5 years.
2. Find the mean of the dataset.
3. Find the median of the dataset.

2. True or false - for the dataset: 13, 13, 15, 15, 16, 16, 16, 17, 18, 22

1. The maximum value of the dataset above is 22. If the analyst makes one mistake and record it as 2.2, then the mistake will change the mean and the median of the dataset. Show calculations.
2. If the variance of a dataset is zero, then all observations must also be zero.
3. The standard deviation of dataset cannot be negative.
4. The variance of a dataset is always positive.
5. The following graph have a negative correlation:

Answer 1

Solution :-

a) Histogram with bin size = 5 years ( We can use any statistics tool to generate histogram like Excel or Numbers)

(Note :- Here X axis shows ages of a group of tourists visiting The Library of Congress and Y axis shows Frequency of that age group)

b) Mean of the dataset can be found via following formula :-

Mean

c) For finding Median first order the numbers from smallest to largest. Then find the middle number.

Here in the dataset Median = 37 years

2. Now for Dataset 13, 13, 15, 15, 16, 16, 16, 17, 18, 22

a) True

Mean of the true dataset (with 22) = 16 years

Median of the true dataset (with 22) = 16.1 years

Mean of the dataset with mistake (with 2.2) = 14.12 years

Median of the dataset with mistake (with 2.2) = 15.5 years

b) False

Variance of the dataset is given as

If all observations are zero(0) than variance is zero.

Also if all observations are same (let's say x) then also variance of that dataset is zero.

If the dataset is 10,10,10,10 ... Mean = 10 and variance = 0.

c) True

Standard Deviation of the dataset is given as

Here we are taking first square and then under-root so it is always wither positive or zeros. Hence standard deviation cannot be negative.

d) False

Variance of the dataset can be given as

As we are taking square , so either variance is positive or zero. For example the variance of the dataset 10,10,10 is 0. And 0 is neither positive nor negative.