In an all boys school, the heights of the student body are normally distributed with a mean of 67 inches and a standard deviation of 5 inches. What percentage of the students are between 59 and 80 inches tall, to the nearest tenth?Statistics Calculator
Question
In an all boys school, the heights of the student body are normally distributed with a mean of 67 inches and a standard deviation of 5 inches. What percentage of the students are between 59 and 80 inches tall, to the nearest tenth?Statistics Calculator
Solution
To solve this problem, we need to use the properties of the normal distribution.
Step 1: Standardize the heights We first convert the heights 59 and 80 inches into z-scores, which are measures of how many standard deviations an element is from the mean. The formula for calculating the z-score is:
Z = (X - μ) / σ
where: X is the value we are standardizing, μ is the mean, and σ is the standard deviation.
For 59 inches: Z1 = (59 - 67) / 5 = -1.6
For 80 inches: Z2 = (80 - 67) / 5 = 2.6
Step 2: Find the probabilities Next, we look up these z-scores in the standard normal distribution table, or use a calculator that can compute probabilities for the standard normal distribution.
The table or calculator gives the probability that a standard normal random variable is less than Z.
For Z1 = -1.6, the probability P(Z < -1.6) is approximately 0.0548, or 5.48%. This is the percentage of students who are shorter than 59 inches.
For Z2 = 2.6, the probability P(Z < 2.6) is approximately 0.9953, or 99.53%. This is the percentage of students who are shorter than 80 inches.
Step 3: Find the percentage of students between 59 and 80 inches To find the percentage of students between 59 and 80 inches, we subtract the percentage of students shorter than 59 inches from the percentage of students shorter than 80 inches.
So, the percentage of students between 59 and 80 inches is approximately 99.53% - 5.48% = 94.05%.
Rounded to the nearest tenth, this is 94.1%.
Similar Questions
For an upcoming government project, you are required to find the average height of the students in Class VIII of a given school. Instead of asking every student, you took a few students as your sample and noted their heights in the table given below. Roll Number Height8012 121.92 cm8045 133.21 cm8053 141.34 cm8099 126.23 cm8125 175.74 cmQuestion 3/3MandatorySamplingWhat is the standard deviation of the sample?22.8423.6719.1921.45
The distribution of heights of 6-year-old girls is approximately normally distributed with a mean of 46.0 inchesand a standard deviation of 2.7 inches. Aliyaah is 6 years old, and her height is 0.96 standard deviation above themean. Her friend Jayne is also 6 years old and is at the 93rd percentile of the height distribution. At whatpercentile is Aliyaah’s height, and how does her height compare to Jayne’s height?(A) Aliyaah’s height is at the 17th percentile of the distribution, and she is shorter than Jayne.(B) Aliyaah’s height is at the 67th percentile of the distribution, and she is shorter than Jayne.(C) Aliyaah’s height is at the 67th percentile of the distribution, and she is taller than Jayne.(D) Aliyaah’s height is at the 83rd percentile of the distribution, and she is shorter than Jayne.(E) Aliyaah’s height is at the 83rd percentile of the distribution, and she is taller than Jayne
Suppose the exam scores on the economics exam are approximately normally distributed.If 15% of the students score below 65 and 10% above 90, then what fraction of the studentsscore (a) below 80 (b) Between 70 and 95?
People's heights are normally distributed, with a mean of 170cm and standard deviation of 10cm. What is the 10th percentile of people's heights (in cm).Please round your result to one decimal place.Note that from the z-distribution table, we know:P(Z<= -1.645) = 5%P(Z<= -1.96) = 2.5%P(Z<= -1.28) = 10%
The heights of a particular breed of horse are normally distributed with a standard deviation of 6.5cm. The mean is unknown but is somewhere between 140cm and 150cm. Which one of the following statements cannot be true?Group of answer choicesThe median of the distribution is 148cmA large horse of this breed could plausibly reach a height of 163cmAbout 5% of horses of this breed have heights outside the range 130cm to 156cm.About 5% of horses of this breed have heights outside the range 142cm to 155cm.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.