Trading volume on the New York Stock Exchange is heaviest...

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?

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?

Key Concepts

Descriptive Statistics

This concept involves summarizing data sets using measures such as the mean (average) and standard deviation (a measure of data dispersion). The mean provides a central value while the standard deviation quantifies how spread out the data points are from that central value. These statistics are often used as estimates for the population parameters when the sample size is limited.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is defined by its mean and standard deviation, and it is widely used in statistics because many real-world phenomena are approximately normally distributed when aggregated. Understanding this distribution is crucial for approximating probabilities and percentiles for various outcomes.

Standard Normal Distribution and Z-Scores

Z-scores represent the number of standard deviations a specific data point is from the mean. By transforming data into z-scores, the values can be compared against the standard normal distribution, which has a mean of 0 and a standard deviation of 1. This transformation is a key step in many statistical calculations, particularly in hypothesis testing and probability estimation.

Probability Calculations using the Z-distribution

Once values are standardized to z-scores, the standard normal distribution can be used to determine the probability of data falling above or below a certain threshold. This process involves finding the area under the curve corresponding to the z-score, which represents the likelihood of a certain outcome occurring.

Quantiles and Critical Values

Quantiles, including percentiles, are cut points dividing the probability distribution into intervals with equal probabilities. Critical values determined from the standard normal distribution are used to identify thresholds, such as the top 5% of outcomes. These values are essential for decision-making and establishing confidence or significance levels in statistical analysis.

Transcript

00:01 So this table right here represent the frequency distribution table for the data we're presented in the problem.

00:07 For the first question, we are to compute the mean and the standard deviation.

00:12 To compute the mean, we'll need the first three columns of the table, that is x, f, and fx.

00:20 Because the mean is equal to the sum of fx divided by the sum of f.

00:25 Sum of fx is given us 2 ,596 divided by the sum of f the frequency which is 13.

00:35 Therefore, our mean is equal to 199 .6923.

00:48 Let's compute the standard deviation.

00:54 The standard deviation is equal to the square root of f times the deviation squared divided by sigma f minus 1 now the deviation squared is equal to the x values minus the mean and we square the result now this column is f times the deviation square which is the deviation square times the column f so the standard deviation is equal to a square root of 8 ,137 divided by 13 minus 1 therefore our standard deviation is equal to 26 .0397.

01:52 The next question, we'll find the probability that trading volumes is less than 180 million shares.

02:00 This is equal to the probability that x is less than 180.

02:05 We need to compute the z's value for the corresponding x and the formula is z is equal to x minus the mean divided by the standard deviation.

02:16 So we can rewrite this as probability that z is less than the x value which is 180 minus the mean 199 .69, tries to run it up to two decimal places.

02:37 Divided by the standard deviation, which is 26 .04.

02:48 So this gives us probability that z is less than negative 0 .76.

02:56 So this probability here is equal to the normal value of negative 0 .76 from the z distribution table...

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