Math Problem Statement

Consider a sample with six observations of 16, 11, 13, 22, 15, and 19. Compute the z-score for each observation. (Leave no cells blank - be certain to enter "0" wherever required. Round your answers to 2 decimal places. Negative values should be indicated by a minus sign.)

Solution

To compute the z-score for each observation in the sample, we use the formula:

z=xμσz = \frac{x - \mu}{\sigma}

Where:

  • xx is the observation,
  • μ\mu is the mean of the sample,
  • σ\sigma is the standard deviation of the sample.

Step 1: Compute the mean (μ\mu) of the sample

The sample is 16,11,13,22,15,1916, 11, 13, 22, 15, 19.

μ=16+11+13+22+15+196=966=16\mu = \frac{16 + 11 + 13 + 22 + 15 + 19}{6} = \frac{96}{6} = 16

Step 2: Compute the standard deviation (σ\sigma)

The standard deviation is computed using the formula for the sample standard deviation:

σ=(xiμ)2n1\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n-1}}

Where:

  • xix_i is each observation,
  • μ\mu is the mean,
  • nn is the number of observations.

Let's calculate it:

  1. (1616)2=0(16 - 16)^2 = 0
  2. (1116)2=25(11 - 16)^2 = 25
  3. (1316)2=9(13 - 16)^2 = 9
  4. (2216)2=36(22 - 16)^2 = 36
  5. (1516)2=1(15 - 16)^2 = 1
  6. (1916)2=9(19 - 16)^2 = 9

Summing these squared differences:

0+25+9+36+1+9=800 + 25 + 9 + 36 + 1 + 9 = 80

Now, compute the standard deviation:

σ=8061=805=16=4\sigma = \sqrt{\frac{80}{6-1}} = \sqrt{\frac{80}{5}} = \sqrt{16} = 4

Step 3: Compute the z-scores

Now that we have μ=16\mu = 16 and σ=4\sigma = 4, we can compute the z-score for each observation:

  1. For x=16x = 16: z=16164=0z = \frac{16 - 16}{4} = 0

  2. For x=11x = 11: z=11164=54=1.25z = \frac{11 - 16}{4} = \frac{-5}{4} = -1.25

  3. For x=13x = 13: z=13164=34=0.75z = \frac{13 - 16}{4} = \frac{-3}{4} = -0.75

  4. For x=22x = 22: z=22164=64=1.50z = \frac{22 - 16}{4} = \frac{6}{4} = 1.50

  5. For x=15x = 15: z=15164=14=0.25z = \frac{15 - 16}{4} = \frac{-1}{4} = -0.25

  6. For x=19x = 19: z=19164=34=0.75z = \frac{19 - 16}{4} = \frac{3}{4} = 0.75

Final z-scores:

  • For x=16x = 16, z-score = 0
  • For x=11x = 11, z-score = -1.25
  • For x=13x = 13, z-score = -0.75
  • For x=22x = 22, z-score = 1.50
  • For x=15x = 15, z-score = -0.25
  • For x=19x = 19, z-score = 0.75

Would you like any more details or have any further questions?


Here are 5 related questions to deepen your understanding:

  1. What is the significance of a z-score in statistics?
  2. How does changing the sample size affect the standard deviation?
  3. Can z-scores be used to compare values from different datasets?
  4. What are the properties of a normal distribution in terms of z-scores?
  5. How does skewness in a dataset affect the z-scores?

Tip: A z-score helps determine how far a data point is from the mean in units of standard deviation.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Statistics
Descriptive Statistics
Z-score
Standard Deviation
Mean

Formulas

z = (x - μ) / σ
Mean (μ) = Σx / n
Standard Deviation (σ) = sqrt(Σ(x_i - μ)^2 / (n - 1))

Theorems

Empirical Rule
68-95-99.7 Rule

Suitable Grade Level

Grades 9-12