The weight of adult green sea urchins are normally distributed with a mean 52g and standard deviation 17.2g?
A. Find the percentage of adult green sea urchins with weighs between 50g and 60g
My answer/work: normalcdf (50, 60, 52, 17.2) = .225, 23%
B. Obtain the percentage of adult green sea urchins with weights above 40g
My answer/work: normalcdf (40, e99, 52, 17.2) = .757, 76%
C. Determine and interpret the 90% percentile for the weights
My answer/work: invnorm: .9, 0,1) =1.28
z-52 + (1.28)*(17.2)
z= 74.04%
I'm a little unsure if I got C correct. As for the rest I'm wondering if I did them correctly. If not, could someone lean me to the right direction? Thank you :)
A. Z = (score-mean)/SD
Z = (50-52)/17.2 = -2/17.2 = -.12
Z = (60-52)/17.2 = 8/17.2 = .47
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z scores.
.0478 +.1808 = .2286 = 23%
B. Similar process.
C. Correct
A. To find the percentage of adult green sea urchins with weights between 50g and 60g, you should use the cumulative distribution function (CDF) by subtracting the area to the left of 50g from the area to the left of 60g.
Using the formula: normalcdf(50, 60, 52, 17.2)
normalcdf(50, 60, 52, 17.2) = 0.342
So, the percentage of adult green sea urchins with weights between 50g and 60g is approximately 34.2%.
B. To find the percentage of adult green sea urchins with weights above 40g, you need to find the area to the right of 40g using the CDF.
Using the formula: normalcdf(40, e99, 52, 17.2)
normalcdf(40, e99, 52, 17.2) = 0.933
So, the percentage of adult green sea urchins with weights above 40g is approximately 93.3%.
C. To determine the 90th percentile for the weights, you need to find the value of the sea urchin weight that corresponds to the cumulative probability of 90%.
Using the inverse normal function: invNorm(0.9, 52, 17.2)
invNorm(0.9, 52, 17.2) = 69.334
Therefore, the 90th percentile for the weights of adult green sea urchins is approximately 69.334g. This means that 90% of the adult green sea urchins have a weight at or below this value.
Let's go through each part of the question to check your answers:
A. Finding the percentage of adult green sea urchins with weights between 50g and 60g:
To find this percentage, you can use the formula for the cumulative distribution function (cdf) of the normal distribution. The formula is:
normalcdf(lower bound, upper bound, mean, standard deviation)
Given that the mean is 52g and the standard deviation is 17.2g, you correctly used the formula as:
normalcdf(50, 60, 52, 17.2)
Calculating this using a calculator or software, you should get a result of approximately 0.225 or 22.5%. Therefore, your answer for part A is incorrect. The correct answer is approximately 22.5%, not 23%.
B. Obtaining the percentage of adult green sea urchins with weights above 40g:
To find this percentage, you need to find the complement of the cumulative distribution function (1 - cdf) of the normal distribution. Using the formula:
1 - normalcdf(lower bound, upper bound, mean, standard deviation)
Given that the mean is 52g and the standard deviation is 17.2g, you correctly used the formula as:
1 - normalcdf(40, e99, 52, 17.2)
The upper bound is set to e99 (a large value close to infinity) to include all weights above 40g. Calculating this using a calculator or software, you should get a result of approximately 0.757 or 75.7%. Therefore, your answer for part B is correct.
C. Determining and interpreting the 90th percentile for the weights:
To find the 90th percentile, you need to use the inverse normal distribution function (invnorm). This function gives you the z-value associated with a given percentile. The formula to find the z-value is:
invnorm(percentile, mean, standard deviation)
Given that you want the 90th percentile (0.9), and the mean is 52g, and the standard deviation is 17.2g, you correctly used the formula as:
invnorm(0.9, 52, 17.2)
Calculating this using a calculator or software, you should get a z-value of approximately 1.28. To find the weight value associated with this z-value, you correctly used the formula:
z = (weight - mean) / standard deviation
Rearranging this formula to solve for weight, you get:
weight = mean + (z * standard deviation)
weight = 52 + (1.28 * 17.2)
weight ≈ 74.036
Therefore, your answer for part C is correct. The 90th percentile for the weights of adult green sea urchins is approximately 74.036g.
To recap:
A. The percentage of adult green sea urchins with weights between 50g and 60g is approximately 22.5%.
B. The percentage of adult green sea urchins with weights above 40g is approximately 75.7%.
C. The 90th percentile for the weights of adult green sea urchins is approximately 74.036g.