Lesson Worksheet: Normal Distribution Mathematics

In this worksheet, we will practice using the normal distribution to calculate probabilities and find unknown variables and parameters.

Q1:

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

Q2:

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

Q3:

For a normally distributed data set with mean 32.1 and standard deviation 2.8, between which two values would you expect 95%‎ 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

Q4:

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

Q5:

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

Q6:

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

Q7:

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

Q8:

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

Q9:

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

Q10:

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

Q11:

Let 𝑋 be a normal random variable. Find 𝑃(𝑋>𝜇+0.71𝜎).

Q12:

Suppose 𝑋 is normally distributed with mean 𝜇 and variance 196. Given that 𝑃(𝑋≤40)=0.0668, find the value of 𝜇.

Q13:

Given a normal random variable 𝑋 such that 𝑃(𝜇−𝑘𝜎≤𝑋≤𝜇+𝑘𝜎)=0.8558, find the value of 𝑘.

Q14:

Let 𝑋 be a random variable which is normally distributed with mean 68 and standard deviation 3. Determine 𝑃(𝑋≥61.7).

Q15:

Let 𝑋 be a random variable which is normally distributed with mean 63 and variance 144. Determine 𝑃(37.56≤𝑋≤57.36).

Q16:

Let 𝑋 be a random variable which is normally distributed with mean 𝜇=75 and standard deviation 𝜎=6. Given that 𝑃(𝑋≥𝑘)=0.9938, find 𝑘.

Q17:

Let 𝑋 be a random variable which is normally distributed with mean 𝜇 and standard deviation 𝜎. Given that 𝑃(𝑋≤72.44)=0.6443 and 𝑃(𝑋≥37.76)=0.9941, calculate the values of 𝜇 and 𝜎.

  • A𝜇=160, 𝜎=35
  • B𝜇=68, 𝜎=12
  • C𝜇=107, 𝜎=14
  • D𝜇=309, 𝜎=94

Q18:

Let 𝑋 be a random variable which is normally distributed with 𝜇=65 and 𝜎=12. Given that 𝑃(𝑋≤𝑘)=0.4013, find 𝑘.

Q19:

Let 𝑋 be a random variable which is normally distributed with mean 60 and standard deviation 5. Determine 𝑃(𝑋≤71).

Q20:

Let 𝑋 be a normal random variable. Find 𝑃(𝜇−0.56𝜎<𝑋<𝜇+1.64𝜎).

Q21:

Let 𝑋 be a normal random variable. Find 𝑃(𝑋<𝜇−1.68𝜎).

Q22:

Let 𝑋 be a random variable which is normally distributed with mean 73 and standard deviation 15. Determine 𝑃(59.05≤𝑋≤110.35).

Q23:

Let 𝑋 be a random variable which is normally distributed with mean 𝜇 and standard deviation 𝜎. Given that 𝑎>0 and 𝑃(𝜇−𝑎𝜎≤𝑋≤𝜇+𝑎𝜎)=2𝑃(0≤𝑋≤1.75), find the value 𝑎.

Q24:

Let 𝑋 be a random variable which is normally distributed with standard deviation 10. Given that 𝑃(𝑋≥69)=0.6554, find the mean of 𝑋.

Q25:

Let 𝑋 be a normal random variable such that 𝑃(𝑋≤93)=0.9821 and 𝜎=10. Calculate 𝜇.

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