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Answer :
a. The mean as 11 grams and the standard deviation as 2 grams.
b. The percent of spiders that weigh between 7 grams and 13 grams is approximately 81%.
c. The percent of spiders that weigh less than 9 grams is approximately 16%.
d. The weight values between 9 and 13 grams include approximately 68% of all the spider weights.
e. The z-score of a spider that weighs 15 grams is 2.
f. The percent of spiders that weigh more than 15 grams is approximately 2.28%.
a. To sketch a normal curve, we start with a horizontal axis representing the weight of the spiders. The vertical axis represents the frequency or probability density of the weights. We label the mean as 11 grams and the standard deviation as 2 grams. Then we mark one, two, and three standard deviations on either side of the mean, which are 9, 13, 15, 7, and 5 grams, respectively.
b. To find the percent of spiders that weigh between 7 grams and 13 grams, we need to calculate the z-scores for these weights. The z-score formula is z = (x - μ) / σ, where x is the weight, μ is the mean, and σ is the standard deviation.
For x = 7, z = (7 - 11) / 2 = -2.
For x = 13, z = (13 - 11) / 2 = 1.
The area under the curve between z = -2 and z = 1 represents the percentage of spiders in this range. Using a standard normal distribution table or a calculator, we find this area to be approximately 81%.
c. To find the percent of spiders that weigh less than 9 grams, we need to calculate the z-score for this weight. For x = 9, z = (9 - 11) / 2 = -1. The area under the curve to the left of z = -1 represents the percentage of spiders that weigh less than 9 grams. Using a standard normal distribution table or a calculator, we find this area to be approximately 16%.
d. We know that approximately 68% of all the spider weights fall within one standard deviation of the mean. Therefore, the weight values between 9 and 13 grams include approximately 68% of all the spider weights.
e. To find the z-score of a spider that weighs 15 grams, we use the z-score formula as z = (x - μ) / σ. For x = 15, μ = 11, and σ = 2, we get z = (15 - 11) / 2 = 2. The z-score of a spider that weighs 15 grams is 2.
f. To find the percent of spiders that weigh more than 15 grams, we need to calculate the area under the curve to the right of z = 2. Using a standard normal distribution table or a calculator, we find this area to be approximately 2.28%. Therefore, approximately 2.28% of all the spiders in the colony weigh more than 15 grams.
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