From statements in J. R. R. Tolkien's "The Fellowship of the Ring" (Prologue I, paragraph 4), I construe that the heights of hobbits are approximately normally distributed with mean 36 inches and standard deviation 4.2 inches. Find the following probabilities (be as accurate as our process allows). A. Find the probability that one hobbit picked at random is no more than 36.7 inches tall. B. Find the probability that a simple random sample of 16 hobbits have a mean height of no more than 36.7 inches tall. C. Find the probability that a simple random sample of 100 hobbits have a mean height of no more than 36.7 inches tall. D. Find the probability that a simple random sample of 400 hobbits have a mean height of no more than 36.7 inches tall.
Mathematics · College · Thu Feb 04 2021
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A. To find the probability that one hobbit picked at random is no more than 36.7 inches tall, we use the Z-score formula for an individual value:
\[ Z = \frac{X - \mu}{\sigma} \]
where: - \( X \) is the value we are measuring (36.7 inches), - \( \mu \) is the mean (36 inches), and - \( \sigma \) is the standard deviation (4.2 inches).
Plugging in the values:
\[ Z = \frac{36.7 - 36}{4.2} = \frac{0.7}{4.2} \approx 0.1667 \]
We now look up this Z-score in the standard normal distribution table (or use a calculator or software that provides the cumulative distribution function for the standard normal distribution). The probability corresponding to a Z-score of 0.1667 is approximately 0.5662. So, the probability of picking a hobbit at random who is no more than 36.7 inches tall is about 0.5662 or 56.62%.
B, C, and D. For a simple random sample of hobbits, we're interested in the distribution of the sample mean, which according to the Central Limit Theorem, the distribution of the sample mean approximates a normal distribution (even if the underlying population distribution is not normal) with mean \( \mu \) (same as the population mean) and standard error \( \sigma_{\bar{x}} \) given by:
\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]
where \( n \) is the sample size.
B. For a sample of 16 hobbits:
\[ \sigma_{\bar{x}} = \frac{4.2}{\sqrt{16}} = \frac{4.2}{4} = 1.05 \]
Now calculate the Z-score for the sample mean:
\[ Z = \frac{\bar{X} - \mu}{\sigma_{\bar{x}}} = \frac{36.7 - 36}{1.05} \approx 0.6667 \]
Looking up a Z-score of 0.6667 gives us a probability of approximately 0.7475. Therefore, the probability that a sample of 16 hobbits has a mean height of no more than 36.7 inches tall is about 0.7475 or 74.75%.
C. For a sample of 100 hobbits:
\[ \sigma_{\bar{x}} = \frac{4.2}{\sqrt{100}} = \frac{4.2}{10} = 0.42 \]
Z-score calculation:
\[ Z = \frac{36.7 - 36}{0.42} \approx 1.6667 \]
The probability for a Z-score of 1.6667 is approximately 0.9525. So the probability is about 0.9525 or 95.25%.
D. For a sample of 400 hobbits:
\[ \sigma_{\bar{x}} = \frac{4.2}{\sqrt{100}} = \frac{4.2}{10} = 0.42 \]
Z-score calculation:
\[ Z = \frac{36.7 - 36}{0.42} \approx 1.6667 \]
The probability for a Z-score of 1.6667 is approximately 0.9525. So the probability is about 0.9525 or 95.25%.
D. For a sample of 400 hobbits:
\[ \sigma_{\bar{x}} = \frac{4.2}{\sqrt{400}} = \frac{4.2}{20} = 0.21 \]
Z-score calculation:
\[ Z = \frac{36.7 - 36}{0.21} \approx 3.3333 \]
For a Z-score of 3.3333, the probability is very close to 1 (almost certain). This is because as the sample size increases, the standard error decreases, making the distribution of the sample mean tighter around the population mean. Thus, it becomes highly probable that the mean height of such a large sample will be close to the population mean.
Extra: Probabilities and distributions are a fundamental concept in statistics, used to determine the likelihood of different outcomes. The normal distribution, which is a bell-shaped curve, is particularly important because many natural phenomena are normally distributed, or approximately so.
The Z-score is a tool used to determine how many standard deviations an element is from the mean. When dealing with a single observation, the standard deviation of the population is used. However, when dealing with samples, particularly large samples, we use the standard error to accommodate the extra variability brought by taking an average rather than an individual measurement.
The Central Limit Theorem is a key concept in statistics as it allows us to make probability statements about the sample mean even when the population distribution is not normal. As the sample size increases, the distribution of the sample mean approaches a normal distribution regardless of the shape of the population distribution, provided the observations are independent and identically distributed with a finite standard deviation.