The average stock price for companies making up the S&P 500 is 30, and the standard deviation i

step 1 This problem is one on normal distribution. We have a mean value of $30, the standard deviation

The average stock price for companies making up the S&P 500...

The average stock price for companies making up the $\mathrm{S\&P} 500$ is $\$ 30,$ and the standard deviation is $\$ 8.20$ (BusinessWeek, Special Annual Issue, Spring 2003 ). Assume the stock prices are normally distributed. a. What is the probability a company will have a stock price of at least $\$ 40 ?$ b. What is the probability a company will have a stock price no higher than $\$ 20 ?$ c. How high does a stock price have to be to put a company in the top $10 \% ?$

The average stock price for companies making up the $\mathrm{S\&P} 500$ is $\$ 30,$ and the standard deviation is $\$ 8.20$ (BusinessWeek, Special Annual Issue, Spring 2003 ). Assume the stock prices are normally distributed.
a. What is the probability a company will have a stock price of at least $\$ 40 ?$
b. What is the probability a company will have a stock price no higher than $\$ 20 ?$
c. How high does a stock price have to be to put a company in the top $10 \% ?$

Key Concepts

Percentiles and Quantiles

Percentiles and quantiles are values that divide a dataset into intervals with equal probabilities. In the context of the normal distribution, they are used to determine thresholds (like the top 10% cutoff) beyond which a certain percentage of observations lie, leveraging the properties of the CDF.

Cumulative Distribution Function (CDF)

The CDF of a normal distribution gives the probability that a random variable is less than or equal to a particular value. It is a crucial tool used to determine the probability associated with a Z-score by integrating the probability density function from negative infinity to the given score.

Mean and Standard Deviation

The mean is the central value or average of the distribution, while the standard deviation measures the spread or dispersion of the data around this mean. In a normal distribution, these two parameters fully describe the shape of the curve.

Normal Distribution

A normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve and is used in statistics to represent real-valued random variables whose distributions are not known.

Z-Score

The Z-score is a standardized score that indicates how many standard deviations an element is from the mean. It transforms any normal distribution into a standard normal distribution, which has a mean of 0 and a standard deviation of 1, making it easier to compute probabilities and percentiles.

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