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Trading volume on the New York Stock Exchange is heaviest during the first half hour (early morning) and last half hour (late afternoon) of the trading day. The early morning trading volumes (millions of shares) for 13 days in January and February are shown here (Barron's, January 23,\(2006 ;\) February 13,\(2006 ;\) and February 27,2006 ). \(\begin{array}{lllll}214 & 163 & 265 & 194 & 180 \\ 202 & 198 & 212 & 201 & \\ 174 & 171 & 211 & 211\end{array}\) The probability distribution of trading volume is approximately normal. a. Compute the mean and standard deviation to use as estimates of the population mean and standard deviation. b. What is the probability that, on a randomly selected day, the early morning trading volume will be less than 180 million shares? c. What is the probability that, on a randomly selected day, the early morning trading volume will exceed 230 million shares? d. How many shares would have to be traded for the early morning trading volume on a particular day to be among the busiest \(5 \%\) of days?

Step by step solution

01

Calculate the Mean

To compute the mean, add all the trading volumes and divide by the number of days. The trading volumes are: 214, 163, 265, 194, 180, 202, 198, 212, 201, 174, 171, 211, 211. The sum is 2396 and there are 13 values.\[\text{Mean} = \frac{2396}{13} \approx 184.31\]

02

Calculate the Standard Deviation

To find the standard deviation, calculate the variance first. Subtract the mean from each data point, square the result, and then find the average of these squares.1. Calculate squared differences from the mean: \((214 - 184.31)^2, (163 - 184.31)^2, \ldots, (211 - 184.31)^2\)2. Average these squared differences:\[\text{Variance} = \frac{1}{13}\sum_{i=1}^{13}(x_i - 184.31)^2 \approx 823.477\]3. Standard deviation is the square root of the variance:\[\text{Standard Deviation} \approx \sqrt{823.477} \approx 28.686\]

03

Probability of Volume Less than 180 Million

We need to find the probability that a day will have less than 180 million trades using the normal distribution. First, convert 180 to a z-score using:\[z = \frac{180 - 184.31}{28.686} \approx -0.150\]Use standard normal distribution tables or software to find the probability: \( P(Z < -0.150) \approx 0.4404 \). Thus, the probability is about 44.04%.

04

Probability of Volume Greater than 230 Million

Convert 230 to a z-score:\[z = \frac{230 - 184.31}{28.686} \approx 1.594\]Use standard normal distribution tables or software to find the probability: \( P(Z > 1.594) \approx 0.0559 \). Thus, the probability is about 5.59%.

05

Calculate Trading Volume for Top 5% Days

To find how many shares would be traded in the top 5%, determine the z-score corresponding to the 95th percentile (as 100% - 5% = 95%). The z-score for 95th percentile is approximately 1.645.Use the z-score formula:\[x = 184.31 + 1.645 \times 28.686 \approx 231.51\]Thus, approximately 231.51 million shares are required for the trading volume to be among the top 5% busiest days.

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Key Concepts

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

Normal Distribution

The normal distribution is a fundamental concept in statistics, characterized by a bell-shaped curve. It is symmetric and centered around the mean. This distribution is important because many random variables in real life, such as heights, test scores, and stock prices, tend to follow this pattern.
In the context of the exercise, the trading volumes are approximately normally distributed. This means that most daily volumes are near the mean, with fewer days having very high or very low volumes. Understanding this pattern helps in predicting and analyzing future data or anomalies in the dataset.
Key features of a normal distribution include:

• Symmetry about the mean
• The mean, median, and mode are all equal
• The total area under the curve is equal to one

Statistical Mean

The statistical mean, often referred to simply as the mean, is a measure of central tendency. It provides an average of a set of numbers and gives us an idea of the "center" of the data.
To determine the mean of the trading volumes shown in the exercise, we add all recorded volumes and divide by the number of observations, which is 13 days in this case. The result is approximately 184.31 million shares.
Knowing the mean is crucial because it helps us summarize the data with a single value, which we can use to make comparisons or identify trends over time.

Standard Deviation

Standard deviation is a statistic that measures the dispersion of a dataset relative to its mean. A low standard deviation indicates that the values tend to be close to the mean, whereas a high standard deviation indicates that the values are spread out over a wider range.
In our exercise, the calculated standard deviation is about 28.686 million shares. This value gives us insight into the consistency of daily trading volumes. A small value would indicate that trading volumes don’t vary much from day to day, while a larger value suggests more variability.
Standard deviation is useful for determining the percentage of data points expected within certain ranges on a normal distribution curve:

• Approximately 68% of data within one standard deviation of the mean
• Approximately 95% within two standard deviations
• Nearly 99.7% within three standard deviations

Z-score

The z-score is a measure that describes a value's relationship to the mean of a group of values, indicating how many standard deviations a data point is from the mean.
For instance, when we calculate the z-score for 180 million shares in our exercise, we find it to be approximately -0.150. This tells us that 180 million is below the mean by about 0.15 standard deviations.
The z-score is crucial for assessing probabilities in a normal distribution because it translates our data into a standard form that can be compared to a standard normal distribution table:

• A z-score of 0 indicates a value exactly at the mean
• Positive z-scores indicate values above the mean
• Negative z-scores suggest values below the mean

This tool allows us to calculate proportions of the dataset that fall above or below specific thresholds, such as determining the likelihood of high trading volumes on particular days.