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Chapter 6: Problem 55

The average birth weight of domestic cats is about 3 ounces. Assume that the distribution of birth weights is Normal with a standard deviation of \(0.4\) ounce. a. Find the birth weight of cats at the 90 th percentile. b. Find the birth weight of cats at the 10 th percentile.

Short Answer

Expert verified
The birth weight at the 90th percentile is approximately 3.512 ounces while the birthweight at the 10th percentile is approximately 2.488 ounces.

Step by step solution

01

Understand the Problem

Given that the average birth weight of cats is about 3 ounces (expected value or mean \(M\)) and standard deviation is 0.4 ounce (\(\sigma\)). The task is to find the 90th and 10th percentile, which means finding the corresponding values that cause 90% and 10% of all possible birth weights to be less. This can be achieved by using the Z-Score formula \(Z = (X - M) / \sigma\).
02

Calculate 90th Percentile

The Z-score corresponding to the 90th percentile is 1.28. By plugging in the given values into the Z-Score formula, and solving for \(X\), the unknown, it yields \(X = Z * \sigma + M\). Substituting the values will result in \(X = 1.28 * 0.4 + 3 = 3.512\) ounces.
03

Calculate 10th Percentile

In a similar manner, the Z-Score corresponding to the 10th percentile is -1.28. Using the Z-Score formula, \(X = Z * \sigma + M\). Substituting the values will result in \(X = -1.28 * 0.4 + 3 = 2.488\) ounces.

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Most popular questions from this chapter

According to National Vital Statistics, the average length of a newborn baby is \(19.5\) inches with a standard deviation of \(0.9\) inch. The distribution of lengths is approximately Normal. Use a table or technology for each question. Include an appropriately labeled and shaded Normal curve for each part. There should be three separate curves. an What is the probability that a baby will have a length of \(20.4\) inches or more? b. What is the probability that a baby will have a length of \(21.4\) inches or more? c. What is the probability that a baby will be between 18 and 21 inches in length?

The undergraduate admission rate at Harvard University is about \(6 \%\) a. Assuming the admission rate is still \(6 \%\), in a sample of 100 applicants to Harvard, what is the probability that exactly 5 will be admitted? Assume that decisions to admit are independent. b. What is the probability that exactly 95 out of 100 applicants will be rejected?

Babies weighing \(5.5\) pounds or less at birth are said to have low birth weights, which can be dangerous. Full-term birth weights for single babies (not twins or triplets or other multiple births) are Normally distributed with a mean of \(7.5\) pounds and a standard deviation of \(1.1\) pounds. a. For one randomly selected full-term single-birth baby, what is the probability that the birth weight is \(5.5\) pounds or less? b. For two randomly selected full-term, single-birth babies, what is the probability that both have birth weights of \(5.5\) pounds or less? c. For 200 random full-term single births, what is the approximate probability that 7 or fewer have low birth weights? d. If 200 independent full-term single-birth babies are born at a hospital, how many would you expect to have birth weights of \(5.5\) pounds of less? Round to the nearest whole number. c. What is the standard deviation for the number of babies out of 200 who weigh \(5.5\) pounds or less? Retain two decimal digits for use in part \(\mathrm{f}\) f. Report the birth weight for full-term single babies (with 200 births) for two standard deviations below the mean and for two standard deviations above the mean. Round both numbers to the nearest whole num- ber. \(\mathrm{g}\). If there were 45 low-birth-weight full-term babies out of 200 , would you be surprised?

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