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Chapter 6: Problem 57

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, and the birth weight at the 10th percentile is approximately \(2.488\) ounces.

Step by step solution

01

Identify the parameters of the distribution

For a normal distribution, we need to know the mean and standard deviation. According to the problem, the mean birth weight of domestic cats is approximately \(3\) ounces, and the standard deviation is \(0.4\) ounces.
02

Calculate the z-scores

Z-scores for the 90th and 10th percentiles can be looked up from a standard normal distribution table or calculated using a statistical software or calculator. The Z score for the 90th percentile is approximately \(1.28\) and for the 10th percentile was about \(-1.28\). This means that a cat at the 90th percentile weighs 1.28 standard deviations more than the average cat while a cat at the 10th percentile weighs 1.28 standard deviations less than the average cat.
03

Apply the formula to find the birth weights

We can find the actual weights at these percentiles using the formula: \(X = 碌 + Z蟽\) where \(碌\) is the mean, \(Z\) is the z-score, and \(蟽\) is the standard deviation. So for 90th percentile: \(X = 3 + 1.28(0.4) = 3.512\) and for 10th percentile: \(X = 3 - 1.28(0.4) = 2.488\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Percentiles
Percentiles are a way of interpreting data points relative to the entire distribution. They divide the data set into 100 equal parts. If a birth weight is at the 90th percentile, it means it weighs more than 90% of the cat population. Conversely, a birth weight at the 10th percentile weighs more than only 10% of cats.

This is useful for understanding how individual data points stand compared to others. In a normal distribution, percentiles can directly relate to z-scores, which provide a standardized way to determine the position of a data point.
Birth Weight
Birth weight is a specific, often normally distributed, measurement that's crucial in understanding the health and development of newborns, in this case, domestic cats. The mean birth weight helps us understand the average size of newborn cats, which is given as 3 ounces in the exercise.
The distribution of birth weights can show how much variability exists around this average.

When the distribution is normal, as assumed here, most cats' birth weights will cluster around the mean with fewer occurrences of very high or low weights. This is inherently important for breeders and veterinarians monitoring the development of cats.
Z-scores
Z-scores are numerical measurements that describe a value's relationship to the mean of a group of values. They are expressed as the number of standard deviations a particular score is from the mean.

Here, the z-score for the 90th percentile is 1.28. This tells us that a cat whose weight is at the 90th percentile is 1.28 standard deviations above the mean. For the 10th percentile, the z-score is -1.28, indicating it's 1.28 standard deviations below the mean.
  • Z-scores help quickly assess how unusual or common a given data point is.
  • They are crucial when comparing values from different normal distributions.
Standard Deviation
Standard deviation is a key statistic that quantifies the amount of variation or dispersion in a set of data values. The standard deviation in this problem is 0.4 ounces.
This tells us how spread out the cat birth weights are around the mean of 3 ounces.

In a normal distribution:
  • About 68% of data lies within one standard deviation of the mean.
  • Approximately 95% falls within two standard deviations.
Thus, a smaller standard deviation indicates that the birth weights are closer to the mean, while a larger standard deviation suggests more variability. This understanding is vital for interpreting the range and likelihood of various birth weights.

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Most popular questions from this chapter

Use the table or technology to find the answer to each question. Include an appropriately labeled sketch of the Normal curve for each part. Shade the appropriate region. A section of the Normal table is provided in the previous exercise. a. Find the area to the left of a \(z\) -score of \(0.92\). b. Find the area to the right of a \(z\) -score of \(0.92\).

According to the Centers of Disease Control and Prevention, \(52 \%\) of U.S. households had no landline and only had cell phone service. Suppose a random sample of 40 U.S. households is taken. a. Find the probability that exactly 20 the households sampled only have cell phone service. b. Find the probability that fewer than 20 households only have cell phone service. c. Find the probability that at most 20 households only have cell phone service. d. Find the probability that between 20 and 23 households only have cell phone service.

According to the Pew Research Center, \(53 \%\) of millennials (those born between 1981 and 1997 ) reported using a library or bookmobile within the last year. Suppose that a random sample of 200 millennials is taken. a. Complete this sentence: We would expect _____ of the sample to have used a library or bookmobile within the last year, give or take ______. b. Would it be surprising to find that 190 of the sample have used a library or bookmobile within the last year? Why or why not?

Make a list of all possible outcomes for gender when a family has two children. Assume that the probability of having a boy is \(0.50\) and the probability of having a girl is also \(0.50\). Find the probability of each outcome in your list.

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 or less? Round to the nearest whole number. e. 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 number. \(\mathrm{g}\). If there were 45 low-birth-weight full-term babies out of 200 , would you be surprised?
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