in a medical examination, 300 people undergo a height measurement and was found out that the average height of all are 150cm. if the standard deviation is 19, find the number of people that fall under the given conditions: a)with the height less than 132cm. b)with height from 140cm to 180cm. c)with height taller than 200cm d)with height from 160cm to 195cm. e) with height from 150cm to 200cm. f)Determine the range of height that includes the critical 60%of the people
Use Z=(X-μ)/σ
where
Z=z-score
X=variable value
μ=mean
σ=standard deviation
Assuming normal distribution, the left-tail normal distribution table gives the probability of the z-score (or less).
Example, part (a):
If N=300 and random sampling, we can assume normal distribution.
μ=150 cm
σ=19 cm
x=132
z=(132-150)/19=-0.9474
P(X<132)=P(Z<-0.9474)=0.1717 (from left-tail normal probability tables)
=>
approximate number of people with heights <132 = 0.1717*300=51.52=52 persons
great
In a medical examination,
300 people undergo a height
measurement and was found
out that the average height of
all are 150cm. If the standard
deviation is 19, find the number of
people that fall under the given
conditions:a.With height less than
132cm
b.With height from 140cm to
180cm
c.With height taller than 200cm
d.With height from 160cm to
195cm
e.With height from 150cm to 200cm
f.Determine the range of height
that includes the central 60%.
g.45% of the people who
undergo the medical examinations
are qualified to be
part of the varsity. What is the
limit height to be part of the varsity?
1.In a medical examination,
300 people undergo a height
measurement and was found
out that the average height of
all are 150cm. If the standard
deviation is 19, find the number of
people that fall under the given
conditions:a.With height less than
132cm
b.With height from 140cm to
180cm
c.With height taller than 200cm
d.With height from 160cm to
195cm
e.With height from 150cm to 200cm
f.Determine the range of height
that includes the central 60%.
g.45% of the people who
undergo the medical examinations
are qualified to be
part of the varsity. What is the
limit height to be part of the varsity?
Give me the answer
a) Oh, that's a low blow for those under 132cm! Let's see. We know the average height is 150cm and the standard deviation is 19. From this information, we can determine the Z-score for a height of 132cm. Using a Z-score table or a handy calculator, we find that the Z-score is approximately -1.05.
Now, we need to find the proportion of people falling below this Z-score. Luckily, we have our trusty Z-tables which tell us that the proportion of people falling below a Z-score of -1.05 is about 0.1469.
Multiplying this proportion by the total number of people (300), we get approximately 44 people who have a height less than 132cm. Bunny hop on over to them!
b) Ah, a range of heights from 140cm to 180cm. Let's calculate the Z-scores for both ends of this range. For 140cm, the Z-score is approximately -0.53, and for 180cm, the Z-score is approximately +1.58.
Using our Z-tables once again, we find that 0.2974 of the population falls below a Z-score of -0.53, and 0.9429 falls below a Z-score of +1.58.
To calculate the proportion falling within this range, we subtract the lower proportion from the higher proportion: 0.9429 - 0.2974 = 0.6455.
Multiplying this proportion by the total number of people (300), we get approximately 194 people in the height range from 140cm to 180cm.
c) Oh my, towering over the rest! For heights taller than 200cm, we need to find the proportion of people falling above a certain Z-score. Since we don't have the actual Z-score for 200cm, we'll go with an approximate value of +2.63.
Consulting our Z-tables, we find that the proportion of people falling below a Z-score of +2.63 is approximately 0.9958.
To calculate the proportion falling above this Z-score, we subtract the proportion below from 1: 1 - 0.9958 = 0.0042.
Multiplying this proportion by the total number of people (300), we get approximately 1 person with a height taller than 200cm. They must be reaching for the stars!
d) Ah, a range of heights from 160cm to 195cm. Let's determine the Z-scores for both ends of the range. For 160cm, the Z-score is approximately -0.53, and for 195cm, the Z-score is approximately +2.05.
Using our Z-tables, we find that 0.2974 of the population falls below a Z-score of -0.53, and 0.9798 falls below a Z-score of +2.05.
Subtracting the lower proportion from the higher proportion, we find that the proportion falling within this range is 0.9798 - 0.2974 = 0.6824.
Multiplying this proportion by the total number of people (300), we get approximately 205 people in the height range from 160cm to 195cm.
e) Ah, a height range from 150cm to 200cm. To determine the proportion falling within this range, we need to calculate the Z-scores for both ends. For 150cm, the Z-score is approximately -0.79, and for 200cm, the Z-score is approximately +2.63.
Using our Z-tables, we find that 0.2857 of the population falls below a Z-score of -0.79, and 0.9958 falls below a Z-score of +2.63.
Subtracting the lower proportion from the higher proportion, we find that the proportion falling within this range is 0.9958 - 0.2857 = 0.7101.
Multiplying this proportion by the total number of people (300), we get approximately 213 people in the height range from 150cm to 200cm.
f) Ah, the critical 60% of the people! To determine the range of height that includes this percentage, we need to find the Z-scores for the lower and upper percentiles. For the lower percentile, we can find the Z-score value when the cumulative proportion is 0.2 (1 - 0.6 = 0.4).
Consulting our Z-tables, we find that the Z-score for a cumulative proportion of 0.2 is approximately -0.84.
For the upper percentile, we need to find the Z-score value when the cumulative proportion is 0.8 (60% + 20% = 80%). The Z-score for a cumulative proportion of 0.8 is approximately +0.84.
Now that we have our Z-scores, we can calculate the corresponding heights. For the lower percentile, we have: -0.84 * 19 (standard deviation) + 150 (average height) ≈ 134.36cm.
And for the upper percentile, we have: +0.84 * 19 (standard deviation) + 150 (average height) ≈ 165.64cm.
Therefore, the range of height that includes the critical 60% of the people is from approximately 134.36cm to 165.64cm. So, if you see people within this range, you know you're in the critical zone! Good luck!
To solve these problems, we will use the concept of the normal distribution. The normal distribution is a bell-shaped curve that represents the distribution of data around the mean. Using this distribution, we can answer the given questions.
First, let's calculate the z-scores for the given height measurements using the formula:
z = (x - μ) / σ
where z is the z-score, x is the individual's height, μ is the mean height, and σ is the standard deviation.
a) To find the number of people with a height less than 132cm, we need to find the probability that an individual has a z-score less than a certain value. In this case, we want the z-score for 132cm. Using the formula, we find:
z = (132 - 150) / 19 = -18 / 19 ≈ -0.947
Now, we can use a standard normal distribution table or a calculator to find the proportion of data below this z-score. For a z-score of -0.947, the table or calculator will give us the value 0.172.
So, approximately 17.2% of the people have a height less than 132cm.
b) To find the number of people with heights ranging from 140cm to 180cm, we need to find the probability that an individual has a z-score between two values.
For 140cm:
z1 = (140 - 150) / 19 = -10 / 19 ≈ -0.526
For 180cm:
z2 = (180 - 150) / 19 = 30 / 19 ≈ 1.579
Again, we can use a standard normal distribution table or a calculator to find the proportion of data between these two z-scores.
Using the table or calculator, we find that the area to the left of z1 is 0.299 and the area to the left of z2 is 0.943.
By subtracting these two probabilities, we get 0.943 - 0.299 = 0.644.
So, approximately 64.4% of the people have heights ranging from 140cm to 180cm.
c) To find the number of people with a height taller than 200cm, we follow a similar process.
z = (200 - 150) / 19 = 50 / 19 ≈ 2.632
Using the table or calculator, we find the area to the left of z (2.632) is 0.995.
To find the proportion of people with heights taller than 200cm, we subtract this probability from 1: 1 - 0.995 = 0.005.
So, approximately 0.5% of the people have a height taller than 200cm.
d) To find the number of people with heights ranging from 160cm to 195cm, we calculate the z-scores for both limits.
For 160cm:
z1 = (160 - 150) / 19 = 10 / 19 ≈ 0.526
For 195cm:
z2 = (195 - 150) / 19 = 45 / 19 ≈ 2.368
Using the table or calculator, we find the area to the left of z1 is 0.701 and the area to the left of z2 is 0.991.
By subtracting these two probabilities, we get 0.991 - 0.701 = 0.29.
So, approximately 29% of the people have heights ranging from 160cm to 195cm.
e) To find the number of people with heights ranging from 150cm to 200cm, we calculate the z-scores for both limits.
For 150cm:
z1 = (150 - 150) / 19 = 0
For 200cm:
z2 = (200 - 150) / 19 = 50 / 19 ≈ 2.632
Using the table or calculator, we find the area to the left of z1 is 0.5 and the area to the left of z2 is 0.995.
By subtracting these two probabilities, we get 0.995 - 0.5 = 0.495.
So, approximately 49.5% of the people have heights ranging from 150cm to 200cm.
f) To determine the range of heights that include the critical 60% of the people, we need to find the z-scores for the two limits using the cumulative distribution function (CDF) of the standard normal distribution.
To find the lower limit, we calculate the z-score corresponding to the 20th percentile (critical 60% leaves 40% on both sides):
Using the table or calculator, we find that the z-score for the 20th percentile is approximately -0.8416.
z1 = -0.8416
To find the upper limit, we calculate the z-score corresponding to the 80th percentile:
Using the table or calculator, we find that the z-score for the 80th percentile is approximately 0.8416.
z2 = 0.8416
Now, we can calculate the corresponding heights for these z-scores:
For the lower limit:
x1 = μ + z1 * σ = 150 + (-0.8416) * 19 = 150 - 15.9984 ≈ 134cm (rounded to the nearest whole number)
For the upper limit:
x2 = μ + z2 * σ = 150 + (0.8416) * 19 = 150 + 15.9984 ≈ 166cm (rounded to the nearest whole number)
So, the range of heights that includes the critical 60% of the people is approximately from 134cm to 166cm.