Since the length of a downhill ski is related to the height of the individuals renting them, it is fair to assume that a normal distribution would describe the length of women’s skis at rental outlets in Colorado. The mean of the distribution is 150 cm and the standard deviation is 10 cm.
a) What is the proportion of women’s ski lengths that are less than 130 cm?
b) Very long skies are expensive and there are not many people who rent them. What is the longest women’s ski a rental shop should carry so that only 2 percent of the costumers will ask to rent a longer ski?
I don't need help solving, but I need help on knowing whether the probability should be taken from a population or a sample and if central limit theorem is applicable or not.
I personally think that the probability should be taken from a population since there is no sample size given. Also, I think that central limit theorem is not applicable because even though the population is assumed to be normally distributed, there is once again, no sample size to evaluate.
I'm pretty sure your right :)
Okay, thank you!
Google davidmlane normal distribution
You are correct that the probability should be taken from the population in this case. Since there is no sample size given and we are talking about the proportion of women’s ski lengths in rental outlets in Colorado as a whole, we would be interested in the probability from the population.
Regarding the applicability of the central limit theorem (CLT), you are also correct that it is not applicable here. The central limit theorem is generally used when considering a sample mean or a sum of random variables from a population that may or may not be normally distributed. It allows us to make inferences about the population using the distribution of the sample mean. However, in this case, we are directly interested in the proportion of ski lengths, not the mean of a sample. Therefore, the central limit theorem is not applicable.
For the specific questions you posed:
a) To find the proportion of women’s ski lengths that are less than 130 cm, you can use the cumulative distribution function (CDF) of the normal distribution. Given the mean (μ = 150 cm) and standard deviation (σ = 10 cm), you can compute the z-score for 130 cm and then use the standard normal distribution table or calculator to find the corresponding proportion.
b) To determine the longest women’s ski a rental shop should carry so that only 2 percent of the customers will ask to rent a longer ski, you need to find the z-score corresponding to the 2nd percentile (or the 98th percentile depending on how the distribution is presented). This z-score can then be used to find the corresponding ski length using the mean and standard deviation provided.
Remember, for both cases, it is important to assume that the lengths of women's skis follow a normal distribution based on the given information.