Worksheet: Normal Distribution

In this worksheet, we will practice performing calculations on data that is normally distributed and using these calculations to make predictions and solve real-life applications.

Q1:

A crop of apples has a mean weight of 105 g and a standard deviation of 3 g. It is assumed that a normal distribution is an appropriate model for this data. What is the approximate probability that a randomly selected apple from the crop has a weight less than 105 g?

Q2:

For the normal distribution shown, approximately what percentage of data points lie in the shaded region?

Q3:

A crop of apples has a mean weight of 105 g and a standard deviation of 3 g. It is assumed that a normal distribution is an appropriate model for this data. What is the approximate probability that a randomly selected apple from the crop has a weight greater than 111 g?

Q4:

For the normal distribution shown, approximately what percent of data points lie in the shaded region?

Q5:

For a normally distributed data set with mean 32.1 and standard deviation 2.8, between which two values would you expect 95%β€Ž of the data set to lie?

  • A29.3 and 34.9
  • B17.5 and 28.7
  • C14.7 and 31.5
  • D26.5 and 37.7
  • E23.7 and 40.5

Q6:

The masses of a population of blackbirds are normally distributed with mean 103 g and standard deviation 11 g.

To the nearest integer, what percentage of blackbirds have masses less than 110 g?

To the nearest tenth, what percentage of blackbirds have masses greater than 124 g?

To the nearest integer, what percentage of blackbirds have masses between 95 g and 120 g?

Q7:

For the normal distribution shown, approximately what percent of data points lie in the shaded region?

Q8:

For a normally distributed data set, approximately what percentage of data points will lie within one standard deviation of the mean?

Q9:

For a normally distributed data set, approximately what percent of data points will lie within two standard deviations of the mean?

Q10:

For a normally distributed data set with mean 32.1 and standard deviation 2.8, between which two values would you expect 68% of the data set to lie?

  • A26.5 and 37.7
  • B23.7 and 40.5
  • C20.3 and 25.9
  • D14.7 and 31.5
  • E29.3 and 34.9

Q11:

For a normally distributed data set with mean 32.1 and standard deviation 2.8, between which two values would you expect 99.7% of the data set to lie?

  • A17.5 and 28.7
  • B29.3 and 34.9
  • C26.5 and 37.7
  • D23.7 and 40.5
  • E14.7 and 31.5

Q12:

A crop of apples has a mean weight of 105 g and a standard deviation of 3 g. It is assumed that a normal distribution is an appropriate model for this data. What is the approximate probability that a randomly selected apple from the crop has a weight between 102 g and 108 g?

Q13:

A crop of apples has a mean weight of 105 g and a standard deviation of 3 g. It is assumed that a normal distribution is an appropriate model for this data. What is the approximate probability that a randomly selected apple from the crop has a weight between 99 g and 111 g?

Q14:

Use tables to find the normal probability corresponding to a 𝑧-score of 2.13.

Q15:

Use tables to find the normal probability corresponding to a 𝑧-score of βˆ’1.73.

Q16:

The heights of a sample of flowers are normally distributed with mean πœ‡ and standard deviation 12. Given that 10.56% of the flowers are shorter that 47 cm, determine πœ‡.

Q17:

Let 𝑋 be a normal random variable. Find 𝑃(𝑋>πœ‡+0.71𝜎).

Q18:

The monthly salaries of workers at a factory are normally distributed with mean 210 pounds and standard deviation 10 pounds. Determine the probability of choosing at random a worker with a salary between 184 and 233 pounds.

Q19:

The lengths of cylinders produced at a factory follow a normal distribution with mean 72 cm and standard deviation 5 cm. A cylinder is acceptable for sale if its length is between 64.4 cm and 73.4 cm. If a random sample of 1,000 cylinders is chosen, how many cylinders would be acceptable for sale?

Q20:

Suppose 𝑋 is normally distributed with mean πœ‡ and variance 196. Given that 𝑃(𝑋≀40)=0.0668, find the value of πœ‡.

Q21:

The marks from a statistics exam are normally distributed with mean πœ‡ and standard deviation 𝜎. What percentage of students got a mark between (πœ‡βˆ’2.27𝜎) and (πœ‡+1.73𝜎)?

Q22:

In a school, the weights of students are normally distributed with mean 66 kg and variance 16 kg2. What percentage of students weigh between 54 kg and 70 kg?

Q23:

In a school with 1000 students, the heights of students are normally distributed with mean 113 cm and standard deviation 5 cm. How many students are shorter than 121 cm?

Q24:

The heights of a group of students follow a normal distribution with a standard deviation of 20 cm. The probability that a student’s height is less than or equal to 180 cm is equal to the probability that a standard normal variable is less than or equal to 2.2. Find the mean height of the group of students.

Q25:

Given a normal random variable 𝑋 such that 𝑃(πœ‡βˆ’π‘˜πœŽβ‰€π‘‹β‰€πœ‡+π‘˜πœŽ)=0.8558, find the value of π‘˜.

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