Lifetime (in years) number of product
10-15 5
15-20 15
20-25 20
25-30 6
30-35 4
A) calculate the mean,median and mode and compare them.
B) calculate their geometric and harmonic means.
C) calculate the upper quartile, the 4th decile and the 64th percentile
D) estimate the number of product whose life is (a)below 22 (b) above
E) determine type of skewness
A) To calculate the mean, median, and mode, follow these steps:
1. Mean:
- Add up all the lifetime values: (10 + 15 + 15 + 15 + 15 + 20 + 20 + 20 + 20 + 25 + 25 + 25 + 25 + 25 + 25 + 30 + 30 + 30 + 30 + 35 + 35 + 35 + 35) = 520
- Divide the sum by the total number of values: 520 / 22 = 23.64 (rounded to two decimal places).
2. Median:
- Arrange the lifetime values in ascending order: 10, 15, 15, 15, 15, 20, 20, 20, 20, 25, 25, 25, 25, 25, 25, 30, 30, 30, 30, 35, 35, 35, 35.
- Since there are 22 values, the median will be the average of the 11th and 12th values: (25 + 25) / 2 = 25.
3. Mode:
- The mode is the value that appears most frequently. In this case, it is 20 since it appears five times.
B) To calculate the geometric and harmonic means, follow these steps:
1. Geometric Mean:
- Multiply all the lifetime values together: 10 * 15 * 15 * 15 * 15 * 20 * 20 * 20 * 20 * 25 * 25 * 25 * 25 * 25 * 25 * 30 * 30 * 30 * 30 * 35 * 35 * 35 * 35 = 6.37e+45.
- Take the 22nd root (since there are 22 values): (6.37e+45)^(1/22) = 23.58 (rounded to two decimal places).
2. Harmonic Mean:
- Calculate the reciprocal of each lifetime value: 1/10, 1/15, 1/15, 1/15, 1/15, 1/20, 1/20, 1/20, 1/20, 1/25, 1/25, 1/25, 1/25, 1/25, 1/25, 1/30, 1/30, 1/30, 1/30, 1/35, 1/35, 1/35, 1/35.
- Add all the reciprocals together: (1/10) + (1/15) + (1/15) + (1/15) + (1/15) + (1/20) + (1/20) + (1/20) + (1/20) + (1/25) + (1/25) + (1/25) + (1/25) + (1/25) + (1/25) + (1/30) + (1/30) + (1/30) + (1/30) + (1/35) + (1/35) + (1/35) + (1/35) = 11.18.
- Divide the total number of values (22) by the sum of the reciprocals: 22 / 11.18 = 1.97 (rounded to two decimal places).
C) To calculate the upper quartile, 4th decile, and 64th percentile, follow these steps:
1. Upper Quartile (Q3):
- Arrange the lifetime values in ascending order: 10, 15, 15, 15, 15, 20, 20, 20, 20, 25, 25, 25, 25, 25, 25, 30, 30, 30, 30, 35, 35, 35, 35.
- Find the position of the upper quartile: (75/100) * 22 = 16.5 (rounded up to 17th position).
- The upper quartile is the value at the 17th position, which is 30.
2. 4th Decile:
- Arrange the lifetime values in ascending order as mentioned above.
- Find the position of the 4th decile: (40/100) * 22 = 8.8 (rounded up to 9th position).
- The 4th decile is the value at the 9th position, which is 20.
3. 64th Percentile:
- Arrange the lifetime values in ascending order as mentioned above.
- Find the position of the 64th percentile: (64/100) * 22 = 14.08 (rounded up to 15th position).
- The 64th percentile is the value at the 15th position, which is 25.
D) To estimate the number of products whose life is (a) below 22 and (b) above 22:
(a) Below 22:
- Count the number of values that are less than 22: 5 (from the first row).
- So, the estimated number of products with a lifetime below 22 is 5.
(b) Above 22:
- Subtract the number of products below 22 from the total number of products: 22 - 5 = 17.
- So, the estimated number of products with a lifetime above 22 is 17.
E) To determine the type of skewness, you need to examine the distribution of the data. In this case, since we don't have the complete data, we can't accurately determine the skewness. Skewness generally refers to whether the data is symmetric, positively skewed (skewed to the right), or negatively skewed (skewed to the left). By visualizing the full data set, you can observe the shape and determine the skewness.