Activity 3.5 Applied Statistics
Today’s consumers are constantly trying to judge the quality of products. But what is quality? How and by whom is quality determined? Some would say the designer creates specifications, which in turn dictate the quality of a product. That quality is also based on the acceptable value of a part within a whole product. Statistics are commonly used in manufacturing processes to control and maintain quality. This activity will allow you to apply statistics in order to analyze and determine the quality of a set of wooded cubes.
In this activity you will collect data and then perform statistical analyses to determine measures of central tendency and variation of the data. You will also represent the data using a histogram.
1. Part of the manufacturing quality control testing for a toy is to measure the depth of a connector piece that must fit into another part. The designed depth is 4.1 cm. Every tenth part produced on the production line is measured. The following data was collected during a two minute production period. 4.1, 4.1, 4.0, 4.1, 3.9, 4.4, 3.9, 4.3, 4.0, 4.2, 4.0, 3.8
a. Calculate each of the following measures of central tendency. Show your work. Mean: ___4.066667__________
Mode: _____4, 4.1 Bimodal________
b. Calculate each of the following measures of variation for the data set. Show your work. A table has been provided to help you calculate the standard deviations. In the table round values in the last two columns to four decimal places. Report the standard deviation statistics to four decimal places. Range: _____0.6________
Standard Deviation of this data: ___.1648__________
Estimated Standard Deviation for all pieces produced: ______.1721_______ X
c. Create a histogram for the data using the grid below. The horizontal axis should display each length measurement from the minimum to maximum recorded lengths. You may choose to begin with a dot plot and then fill in the bars. Be sure to label your axes.
d. Is the data normally distributed? Justify your answer.
2. Use the mean and (sample) standard deviation to make predictions about the spread of the depth of all of the connector pieces produced in this production run.
a. Calculate each of the following. Round answers to the thousandth of a cm: µ + σ ___________4.2388_____________
µ ‒ σ ___________3.8946_____________
µ + 2σ __________4.4109_____________
µ ‒ 2σ __________3.7225______________
b. Write an inequality that represents all of the data values, X, that fall between the values of µ ± σ. 4.2388 > x > 4.0667
c. Theoretically, if the data is normally distributed, what percentage of the samples should fall within the 1 standard deviation of the mean? 68%
d. What percentage of the data values fall within the limits of µ - σ < X < µ + σ where X is the depth of the connector piece? Note that µ - σ < X < µ + σ is referred to as a compound inequality. Show your work. Does this agree with the theoretical percentage? If not, explain.
e. Write an inequality that represents all of the data values, X, that fall between the values of µ ± 2σ.
f. Theoretically, if the data is normally distributed, what percentage of the samples should fall within one standard deviations of the mean?
g. What percentage of the data values fall within the limits of µ - 2σ < X < µ + 2σ where X is the depth of the connector piece? Note that µ -2 σ < X < µ + 2σ i is referred to as a compound inequality. Show your work. Does this agree with the...
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