The heights in centimetres of men in a sample selected at random may be modelled by a normal distribution with mean 180.5 and standard deviation 11.3.
1. Choose the option that is closest to the value above which 30% of the heights of men will lie.
Options for Question1
A 174.57 B 180.32 C 180.50 D 186.43 E 198.33 F 200.50
2. Choose the option that is closest to a range of values,symmetric
about the mean, within which the heights of approximately 80% of
men will lie.
Options for Question 2
A (157.0, 194.0) B (159.2, 201.8) C (167.8, 193.2) D (163.0, 198.0) E (166.0, 195.0) F (168.7, 192.2)
3. Choose the option that is closest to the percentage of heights of men below 170 cm.
Options for Question 3
A 17% B 18% C 19% D 40% E 41% F 42%
are those correct?
1. is D
2. is E
3. is B ?
Regards.
1. right!
2. right!
3. Right!
To find the answers to these questions, we need to use the concept of the standard normal distribution. This distribution has a mean of 0 and a standard deviation of 1. We can standardize the data using the formula:
Z = (X - μ) / σ
Where:
Z - standardized value
X - original value
μ - mean
σ - standard deviation
To answer the questions:
1. To find the value above which 30% of the heights of men will lie, we need to find the corresponding Z-score using the standard normal distribution table or a calculator. The Z-score corresponding to 30% is approximately -0.52 (rounded to two decimal places). We can then use the formula to find the corresponding height value:
Z = (X - μ) / σ
-0.52 = (X - 180.5) / 11.3
Solving for X, we get:
X ≈ -0.52 * 11.3 + 180.5 ≈ 174.57
So the correct answer is A: 174.57
2. To find a range of values within which approximately 80% of the heights of men will lie, we need to find the Z-score corresponding to 10% on each tail of the distribution. The Z-score corresponding to 10% is approximately -1.28. We can then use the formula to find the corresponding height values:
Z = (X - μ) / σ
-1.28 = (X - 180.5) / 11.3
Solving for X, we get:
X ≈ -1.28 * 11.3 + 180.5 ≈ 165.46
Thus, the lower value of the range is approximately 165.46. Since the range is symmetric about the mean, the upper value will be the same but with a positive sign.
So the correct answer is E: (166.0, 195.0)
3. To find the percentage of heights of men below 170cm, we need to calculate the Z-score for this value and then use the standard normal distribution table or calculator to find the corresponding percentage.
Z = (X - μ) / σ
Z = (170 - 180.5) / 11.3 ≈ -0.93
Using the Z-score table or calculator, we find that the percentage corresponding to a Z-score of -0.93 is approximately 17%.
So the correct answer is A: 17%
Therefore, the correct answers are:
1. A, 2. E, 3. A