The Clothing Buyer For A Chain Of Department Stores Wants To
The Clothing Buyer For A Chain Of Department Stores Wants to Be Sure T
The clothing buyer for a chain of department stores wants to be sure the store orders the right mix of sizes. As part of a survey, she measured the height (in inches) of men who bought suits at this store. Her software reported the confidence interval shown below. Complete parts a through d. (a) Explain carefully what the software output means.
A. She is 5% confident that the mean height of men who visit this store lies between about 71.5 and 73.9 inches.
B. She is 99% confident that the mean height of men who visit this store lies between about 71.5 and 73.9 inches.
C. She is 95% confident that the mean height of men who visit this store lies between about 71.5 and 73.9 inches.
D. She is 95% confident that the mean height of men who visit this store does not lie between 71.5 and 73.9 inches. Your answer is correct. With information given I am not understanding how to get this?
(b) What's the margin of error for this interval? ME = ____ inches (Round to one decimal place as needed.)
Paper For Above instruction
The given scenario involves a clothing buyer at a chain of department stores who wishes to understand customer demographics, specifically focusing on men's heights to ensure appropriate sizing and inventory management. The survey conducted provided a confidence interval for the mean height of men who purchase suits at the store. Precise interpretation of this confidence interval and calculation of the margin of error (ME) are essential to make informed decisions. This paper aims to interpret the software output correctly and determine the margin of error with relevant statistical principles.
Understanding the Confidence Interval
A confidence interval provides a range of values within which the true population parameter (here, the mean height of men buying suits) is estimated to lie with a specified level of confidence, commonly 95%. The software reported a confidence interval from approximately 71.5 to 73.9 inches. The critical step is to interpret what this interval means in context.
Option A suggests a 5% confidence level, implying a confidence that the true mean height lies within the interval with only 95% confidence, which is inconsistent with standard statistical interpretation. Typically, confidence intervals are expressed at 95%, 99%, etc., levels, matching the confidence level mentioned in the options.
Option B claims a 99% confidence level. However, the interval provided (between about 71.5 and 73.9 inches) appears more consistent with a 95% confidence interval, given the typical width of such intervals and the context expressed.
Option C states that she is 95% confident that the true mean height lies within the interval of about 71.5 to 73.9 inches. This aligns with standard statistical practice, where the confidence level is 95%, and the interval reflects this certainty.
Option D incorrectly implies that the confidence level pertains to the true mean not lying within the interval, which is a misinterpretation of confidence intervals. The correct interpretation is that we are confident, at the specified level (95%), that the true mean lies within this range.
Therefore, the best understanding is option C: the woman is 95% confident that the mean height of men who visit the store is between approximately 71.5 and 73.9 inches.
Calculating the Margin of Error
The margin of error (ME) is a measure of the precision of the estimate provided by the confidence interval. It is the distance from the point estimate (sample mean) to either end of the interval.
The confidence interval is expressed as:
Interval = sample mean ± margin of error
Given the interval from approximately 71.5 inches to 73.9 inches, the midpoint (which is the sample mean) can be calculated as:
Mean = (Lower bound + Upper bound) / 2 = (71.5 + 73.9) / 2 = 72.7 inches
The margin of error is equal to:
ME = Upper bound – sample mean = 73.9 – 72.7 = 1.2 inches
Similarly, confirming with the lower bound:
72.7 – 71.5 = 1.2 inches
Therefore, the margin of error is approximately 1.2 inches. Rounded to one decimal place, the margin of error is 1.2 inches.
Conclusion
In conclusion, the confidence interval suggests that with 95% confidence, the true average height of men buying suits at this store lies between 71.5 inches and 73.9 inches. The margin of error associated with this estimate is approximately 1.2 inches. These insights can assist the clothing buyer to make informed decisions regarding the store's inventory sizing, ensuring the store stocks suitable sizes for the target demographic.
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