Math Problem Statement

heckout times. The checkout times​ (in minutes) for 12 randomly selected customers at a large supermarket during the​ store’s busiest time​ are: 5.3 5.3 5.5 9.4 12.7 9.9 6.8 8.9 9.3 8.9 13.1 13.8 find the mean and standard deviation

Solution

To find the mean and standard deviation of the checkout times, we can follow these steps:

1. Calculate the Mean:

The mean (average) is the sum of all values divided by the number of values.

Mean=Sum of all checkout timesNumber of customers\text{Mean} = \frac{\text{Sum of all checkout times}}{\text{Number of customers}}

Given the checkout times: 5.3,5.3,5.5,9.4,12.7,9.9,6.8,8.9,9.3,8.9,13.1,13.85.3, 5.3, 5.5, 9.4, 12.7, 9.9, 6.8, 8.9, 9.3, 8.9, 13.1, 13.8

Sum of the checkout times: 5.3+5.3+5.5+9.4+12.7+9.9+6.8+8.9+9.3+8.9+13.1+13.8=108.05.3 + 5.3 + 5.5 + 9.4 + 12.7 + 9.9 + 6.8 + 8.9 + 9.3 + 8.9 + 13.1 + 13.8 = 108.0

Number of customers = 12

So, the mean is: Mean=108.012=9.0minutes\text{Mean} = \frac{108.0}{12} = 9.0 \, \text{minutes}

2. Calculate the Standard Deviation:

The standard deviation measures how spread out the numbers are. First, we need to calculate the variance, and then take the square root of it.

Variance formula: Variance=(xiμ)2n\text{Variance} = \frac{\sum (x_i - \mu)^2}{n} Where:

  • xix_i is each individual checkout time
  • μ\mu is the mean (9.0)
  • nn is the number of data points (12)

Step-by-step:

  • Subtract the mean from each value, square the result, and sum them up:

(5.39.0)2=(3.7)2=13.69(5.3 - 9.0)^2 = (-3.7)^2 = 13.69 (5.39.0)2=(3.7)2=13.69(5.3 - 9.0)^2 = (-3.7)^2 = 13.69 (5.59.0)2=(3.5)2=12.25(5.5 - 9.0)^2 = (-3.5)^2 = 12.25 (9.49.0)2=(0.4)2=0.16(9.4 - 9.0)^2 = (0.4)^2 = 0.16 (12.79.0)2=(3.7)2=13.69(12.7 - 9.0)^2 = (3.7)^2 = 13.69 (9.99.0)2=(0.9)2=0.81(9.9 - 9.0)^2 = (0.9)^2 = 0.81 (6.89.0)2=(2.2)2=4.84(6.8 - 9.0)^2 = (-2.2)^2 = 4.84 (8.99.0)2=(0.1)2=0.01(8.9 - 9.0)^2 = (-0.1)^2 = 0.01 (9.39.0)2=(0.3)2=0.09(9.3 - 9.0)^2 = (0.3)^2 = 0.09 (8.99.0)2=(0.1)2=0.01(8.9 - 9.0)^2 = (-0.1)^2 = 0.01 (13.19.0)2=(4.1)2=16.81(13.1 - 9.0)^2 = (4.1)^2 = 16.81 (13.89.0)2=(4.8)2=23.04(13.8 - 9.0)^2 = (4.8)^2 = 23.04

Now, sum up all the squared differences: 13.69+13.69+12.25+0.16+13.69+0.81+4.84+0.01+0.09+0.01+16.81+23.04=95.3913.69 + 13.69 + 12.25 + 0.16 + 13.69 + 0.81 + 4.84 + 0.01 + 0.09 + 0.01 + 16.81 + 23.04 = 95.39

Variance: Variance=95.3912=7.95\text{Variance} = \frac{95.39}{12} = 7.95

Finally, take the square root to find the standard deviation: Standard Deviation=7.952.82minutes\text{Standard Deviation} = \sqrt{7.95} \approx 2.82 \, \text{minutes}

Final Results:

  • Mean = 9.0 minutes
  • Standard Deviation ≈ 2.82 minutes

Would you like a more detailed breakdown, or any clarifications? Here are a few follow-up questions to dive deeper:

  1. How can the standard deviation help in understanding customer behavior during checkout?
  2. What is the importance of the mean in determining the efficiency of checkout times?
  3. How can we visualize the distribution of these checkout times?
  4. How would adding more customers affect the mean and standard deviation?
  5. What does a high standard deviation indicate in a set of data?

Tip: A small standard deviation indicates that the data points are close to the mean, while a large one suggests more variability.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Statistics
Mean
Standard Deviation
Variance

Formulas

Mean = (Sum of all values) / (Number of values)
Variance = Sum of squared differences from the mean / Number of values
Standard Deviation = Square root of Variance

Theorems

Basic Properties of Mean and Standard Deviation

Suitable Grade Level

Grades 9-11