Question

Denim Lengths Lab - Use the Normal Probability Distribution to make decisions about a population. Purpose: In clothing production, sizes are based on the average height for females and males. This lab will investigate how a manufacturer might use statistics to make decisions about their manufacturing process. Part 1: Working the female heights – sample data (24 pts) Female Height Data (inches) 62 67 67 67 70 69 63 59 61 62 66 55 64 71 72 69 67 66 64 62 63 68 72 72 65 60 58 64 65 65 64 70 Using technology (Calculator or Desmos) find the mean and standard deviation of the Female Height Data set. Round your values to the first decimal place. Note, the data is not paired, it is written in two columns to save space. Double check to make sure you have typed your data in correctly before proceeding. Find the sample mean and standard deviation. Report those values below. 1. Round these values to the first decimal place. Mean _________ Standard Deviation _______ 2. Using technology (Calculator or Desmos) answer the following questions. Round to the first decimal place. Note the answers should be percent’s. a) What percent of female adults are less than 6 feet (72 inches)? _______ b) What percent of female adults are at least 5 feet (60 inches)? _______ c) What percent of female adult heights are between 60 inches and 72 inches? ______ 3. Practicality makes it impossible to make blue jeans in all sizes. Find the heights corresponding to the following percentages. These are the heights of the shortest and tallest females who may have difficulty finding blue jeans corresponding to their height. You will use the invnorm function of you calculator or the inverse normal distribution on Desmos. Refer to Chapter 6 Section 2 video to see an example of this question. (Round to the nearest inch) a) The bottom 8% = ¬¬¬_________ b) The upper 6% = ¬¬¬¬¬¬¬¬¬¬¬¬________ 4. Use the values in #3 to find the percentage of female adults whose heights are between the lowest 8% and the upper 6%. (Round to first decimal place) _____________ Part 2: Working with Male Heights Now you will answer similar question but will use data for Male heights. (24 pts) Male Height Data (inches) 71 70 60 65 76 74 76 67 75 72 69 65 73 68 73 77 76 71 65 77 72 77 73 64 72 63 70 65 74 71 67 63 Assuming the data is normally distributed, use the techniques from the previous question to find the sample mean and sample standard deviation for the data (Rounded to the first decimal place). Note, the data is not paired, it is written in two columns to save space. Find the sample mean and standard deviation. Report those values below. Round these values to the first decimal place. 5. Mean = ___________ Standard deviation= _____________ 6. Use the techniques from the previous question to answer the questions below. (round the following answers to the first decimal place). a) What percent of male adults are less than 7 feet (84 inches)? ___________ b) What percent of male adults are at least 5 feet 5 inches (65 inches)? ________ c) What percent of male adult heights are between 65 inches and 84 inches? __________ 7. Practicality makes it impossible to make blue jeans in all sizes. Find the heights corresponding to the following percentages. These are the heights of the shortest and tallest males who will have blue jeans corresponding to their heights. You will use the invnorm function of you calculator or the inverse normal distribution on Desmos. (Round to the nearest inch) a) The bottom 8% = _________ b) The upper 6% = _________ 8. Find the percentage of male adults whose heights are between the lowest 8% and the upper 6%. (Round to first decimal place) _____________ Part 3: Compare/contrast 9. Which gender would have the highest probability of finding jeans to fit their height? Explain your answer.

Answer 1

1. Round these values to the first decimal place. Mean 65.2 inches Standard Deviation 5.1 inches.

2. Using technology (Calculator or Desmos) answer the following questions. Round to the first decimal place. Note the answers should be percent’s. a) What percent of female adults are less than 6 feet (72 inches)? 95.8%; b) What percent of female adults are at least 5 feet (60 inches)? 30.0%; c) What percent of female adult heights are between 60 inches and 72 inches? 93.3%.

3. Practicality makes it impossible to make blue jeans in all sizes. Find the heights corresponding to the following percentages. These are the heights of the shortest and tallest females who may have difficulty finding blue jeans corresponding to their height. You will use the invnorm function of you calculator or the inverse normal distribution on Desmos. Refer to Chapter 6 Section 2 video to see an example of this question. (Round to the nearest inch) a) The bottom 8% = ¬¬¬59 inches; b) The upper 6% = ¬¬¬¬¬¬¬¬¬¬¬¬72 inches.

4. Use the values in #3 to find the percentage of female adults whose heights are between the lowest 8% and the upper 6%. (Round to first decimal place) 87.5% Part 2: Working with Male Heights Now you will answer similar question but will use data for Male heights. (24 pts) Male Height Data (inches) 71 70 60 65 76 74 76 67 75 72 69 65 73 68 73 77 76 71 65 77 72 77 73 64 72 63 70 65 74 71 67 63 Assuming the data is normally distributed, use the techniques from the previous question to find the sample mean and sample standard deviation for the data (Rounded to the first decimal place). Note, the data is not paired, it is written in two columns to save space. Find the sample mean and standard deviation. Report those values below. Round these values to the first decimal place.

5. Mean = 70.3 inches Standard deviation= 4.9 inches.

6. Use the techniques from the previous question to answer the questions below. (round the following answers to the first decimal place). a) What percent of male adults are less than 7 feet (84 inches)? 99.9%; b) What percent of male adults are at least 5 feet 5 inches (65 inches)? 40.8%; c) What percent of male adult heights are between 65 inches and 84 inches? 91.7%.

7. Practicality makes it impossible to make blue jeans in all sizes. Find the heights corresponding to the following percentages. These are the heights of the shortest and tallest males who will have blue jeans corresponding to their heights. You will use the invnorm function of you calculator or the inverse normal distribution on Desmos. (Round to the nearest inch) a) The bottom 8% = 62 inches; b) The upper 6% = 78 inches.

8. Find the percentage of male adults whose heights are between the lowest 8% and the upper 6%. (Round to first decimal place) 87.5% Part 3: Compare/contrast.

9. Female jeans would have a higher probability of fitting due to a smaller standard deviation, indicating less variability in female heights.

Step-by-step explanation:

In Part 1, the mean (µ) and standard deviation (σ) for female heights were calculated. Using the normal distribution, we found that a) approximately 95.8% of females are less than 6 feet (72 inches), b) around 30.0% are at least 5 feet (60 inches), and c) about 93.3% fall between 60 and 72 inches. To determine practical sizing, we found the heights corresponding to the bottom 8% (a) and upper 6% (b), which were approximately 59 inches and 72 inches, respectively. The percentage of females in this height range is 87.5% (4).

In Part 2, similar calculations were performed for male heights. The mean (µ) and standard deviation (σ) were found to be 70.3 inches and 4.9 inches, respectively. a) About 99.9% of males are less than 7 feet (84 inches), b) roughly 40.8% are at least 5 feet 5 inches (65 inches), and c) approximately 91.7% fall between 65 and 84 inches. Heights corresponding to the bottom 8% (a) and upper 6% (b) were found to be 62 inches and 78 inches. The percentage of males in this height range is 87.5% (8).

In conclusion, female jeans would have a higher probability of fitting due to the smaller standard deviation, indicating less variability in female heights compared to males (9).