satistics math maria

Question

58 randomly selected students were asked the number of pairs of shoes they have. Let X represent the number of pairs of shoes. The results are as follows:

# of Pairs of Shoes	4	5	6	7	8	9	10	11	12	13	14	15
Frequency	3	3	6	9	7	3	4	5	1	3	6	8

Round all your answers to 4 decimal places where possible.

The mean is: 

The median is: The sample standard deviation is: 

The first quartile is: 

The third quartile is: 

What percent of the respondents have at least 4 pairs of Shoes? 

Answer 1

To calculate the mean, median, sample standard deviation, and quartiles for the given data, we need to use the frequency distribution and perform the necessary calculations.

First, let's calculate the mean.

Mean = Σ(X * f) / Σf

X = number of pairs of shoes

f = frequency

Mean = [(4 * 3) + (5 * 3) + (6 * 6) + (7 * 9) + (8 * 7) + (9 * 3) + (10 * 4) + (11 * 5) + (12 * 1) + (13 * 3) + (14 * 6) + (15 * 8)] / 58

Mean = 366 / 58

Mean ≈ 6.3103

Now, let's find the median. Since there are 58 students, the median will be the average of the 29th and 30th values when the data is sorted in ascending order. The cumulative frequency shows that the median lies in the 7 shoe category, which has a cumulative frequency of 29 + 9 = 38.

Median ≈ 7

Next, let's calculate the sample standard deviation.

Variance = Σ[(X - Mean)² * f] / (Σf - 1)

Variance = [(3 * (4 - 6.3103)²) + (3 * (5 - 6.3103)²) + (6 * (6 - 6.3103)²) + (9 * (7 - 6.3103)²) + (7 * (8 - 6.3103)²) + (3 * (9 - 6.3103)²) + (4 * (10 - 6.3103)²) + (5 * (11 - 6.3103)²) + (1 * (12 - 6.3103)²) + (3 * (13 - 6.3103)²) + (6 * (14 - 6.3103)²) + (8 * (15 - 6.3103)²)] / (58 - 1)

Variance ≈ 10.7196

Sample standard deviation = √Variance

Sample standard deviation ≈ √10.7196

Sample standard deviation ≈ 3.2732

Now, let's find the first and third quartiles.

Q1 is the value that has 25% of the data below it, which corresponds to the 14.5th value in the sorted dataset.

Q3 is the value that has 75% of the data below it, which corresponds to the 43.5th value in the sorted dataset.

Based on the cumulative frequency, Q1 lies in the 6 shoe category, and Q3 lies in the 14 shoe category.

Q1 ≈ 6

Q3 ≈ 14

Finally, let's find the percentage of respondents with at least 4 pairs of shoes. Since the minimum number of pairs of shoes in the data is 4, all 58 respondents have at least 4 pairs of shoes.

Percentage = (58 / 58) * 100

Percentage = 100%

Results:

Mean ≈ 6.3103

Median ≈ 7

Sample standard deviation ≈ 3.2732

Q1 ≈ 6

Q3 ≈ 14

Percentage of respondents with at least 4 pairs of shoes = 100%