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Heights of 10 year olds, regardless of gender, closely follow a normal distribution with mean 55 inches and standard deviation 6 inches. (a) What fraction of 10 year olds are taller than 76 inches? (b) If there are 2,00010 year olds entering Six Flags Magic Mountain in a single day, then compute the expected number of 10 year olds who are at least 76 inches tall. (You may assume the heights of the 10-year olds are independent.) (c) Using the binomial distribution, compute the probability that 0 of the 2,00010 year olds will be at least 76 inches tall. (d) The number of 10 year olds who enter Six Flags Magic Mountain and are at least 76 inches tall in a given day follows a Poisson distribution with mean equal to the value found in part (b). Use the Poisson distribution to identify the probability no 10 year old will enter the park who is 76 inches or taller.

Step by step solution

01

Understanding the Z-Score

To find the fraction of 10-year-olds taller than 76 inches, first, we need to calculate the Z-score. The Z-score formula is \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the value we are investigating, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. Here, \( X = 76 \), \( \mu = 55 \), and \( \sigma = 6 \).

02

Calculation of Z-Score

Now, substitute the values into the Z-score formula: \[ Z = \frac{76 - 55}{6} = \frac{21}{6} = 3.5 \]

03

Finding the Fraction Using Z-Table

Using the Z-table, find the probability that corresponds to \( Z = 3.5 \). The Z-table gives the probability that a value is less than a certain Z-score. For \( Z = 3.5 \), this probability is very close to 1 (specifically, 0.99977 as found in most Z-tables). Therefore, the fraction taller than 76 inches is \( 1 - 0.99977 = 0.00023 \).

04

Calculating the Expected Number

Now calculate the expected number of 10-year-olds who are at least 76 inches tall out of the 2,000 children. Multiply the total number of children by the fraction found in Step 2: \[ 2000 \times 0.00023 = 0.46 \] Since we can't have a fraction of a person, we round to the nearest whole value, which is 0.

05

Using Binomial Distribution

For the binomial probability that 0 of the 2,000 children are at least 76 inches tall, use the binomial formula \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( n = 2000 \), \( k = 0 \), and \( p = 0.00023 \). Thus: \[ P(X = 0) = \binom{2000}{0} (0.00023)^0 (1-0.00023)^{2000} \approx (0.99977)^{2000} \] Calculate \( (0.99977)^{2000} \) to get approximately 0.631. This is the probability that none of the children are 76 inches or taller.

06

Using Poisson Distribution

For the Poisson distribution, use the formula \( P(X=k) = \frac{{e^{-\lambda} \lambda^{k}}}{k!} \), where \( \lambda = 0.46 \) (from Step 3) and \( k = 0 \). Thus: \[ P(X=0) = \frac{e^{-0.46} \times 0.46^0}{0!} = e^{-0.46} \] Calculate to get \( e^{-0.46} \approx 0.630 \). This is the probability that no 10-year-old will enter the park being 76 inches or taller.

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Key Concepts

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

Z-Score

The Z-score is a statistical concept that helps assess how far a particular data point is from the mean of a distribution, measured in units of standard deviation. In the context of the given exercise, the Z-score is used to determine what fraction of 10-year-olds are taller than 76 inches given that their heights are normally distributed.To calculate the Z-score, you'll use the formula: \[ Z = \frac{X - \mu}{\sigma} \]where:

• \( X \) is the value you're investigating
• \( \mu \) is the mean of the distribution
• \( \sigma \) is the standard deviation

Substituting the values from the exercise, where \( X = 76 \), \( \mu = 55 \), and \( \sigma = 6 \), the Z-score calculation becomes:\[ Z = \frac{76 - 55}{6} = 3.5 \]This calculates how many standard deviations 76 inches is above the mean. A Z-score of 3.5 is considerably high, which means that being this tall is unusual among 10-year-olds.

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood of a value taking one of two independent states and is determined by the binomial formula. In this specific exercise, it is used to calculate the probability that zero 10-year-olds out of 2,000 are at least 76 inches tall.The formula for the binomial probability is:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where:

• \( n \) is the number of trials (in this case, 2000)
• \( k \) is the number of successful outcomes (here, 0)
• \( p \) is the probability of success on a single trial (from the Z-score calculation, \( p = 0.00023 \))

Here, you calculate the probability of zero successes by computing:\[ P(X = 0) = (0.99977)^{2000} \]This showcases how the binomial distribution is handy for categorical event outcomes, like determining how unlikely it is for none out of 2,000 children to be exceptionally tall.

Poisson Distribution

The Poisson distribution is another probability distribution used for counting the number of events happening within a fixed interval. This distribution is particularly useful when these events occur with a known constant mean rate and independently of the time since the last event.In this exercise, the Poisson distribution is employed to calculate the probability of no 10-year-old reaching a height of 76 inches or more, based on the mean value computed earlier (\( \lambda = 0.46 \)). The formula is:\[ P(X=k) = \frac{{e^{-\lambda} \lambda^{k}}}{k!} \]where:

• \( \lambda \) represents the average rate (or number of occurrences)
• \( k \) is the number of occurrences (zero, in this case)

For this problem, the calculation is:\[ P(X=0) = e^{-0.46} \]This computes to approximately 0.630. It indicates the probability of witnessing none of the children being at least 76 inches tall on a given day at the park.

Probability Calculation

Probability calculations are key in understanding the likelihood of various outcomes under realistic scenarios. They're the backbone for many statistical analyses from science to business applications, and they provide a measure of how certain we can be about future events. In statistical exercises such as the one provided, several types of probability calculations are used:

• **Z-table Probability:** After calculating a Z-score, use a Z-table to find the probability of data points lying below or above this score. This gives insight into how common or rare a specific data outcome is.
• **Expected Value:** Multiply possible outcomes by their probabilities and sum them for an average result. Here, the expected number of children reaching a height of at least 76 inches is calculated using this logic.
• **Discrete Distributions:** Tools like binomial and Poisson distributions come in handy when calculating outcomes for categorical (binomial) and count data (Poisson).

Probability calculations provide the fuel for informed decision-making by quantifying uncertainty and predicting outcomes over time. No matter the field of study, these calculations help decode randomness into actionable data insights.