Un importante aeropuerto ha contratado recientemente al...

Un importante aeropuerto ha contratado recientemente al consultor Juan Cadaqués para estudiar el problema de los retrasos en el tráfico aéreo. Éste ha anotado el número de minutos de retraso de una muestra de vuelos en la siguiente tabla:$$ \begin{array}{llccccc} \hline \begin{array}{l} \text { Minutos } \\ \text { de retaso } \end{array} & 0<10 & 10<20 & 20<30 & 30<40 & 40<50 & 50<60 \\ \hline \begin{array}{l} \text { Número } \\ \text { de vueles } \end{array} & 30 & 25 & 13 & 6 & 5 & 4 \\ \hline \end{array} $$ a) Estime el nümero medio de minutos de retraso. b) Estime la varianza y la desviación típica muestrales.

Un importante aeropuerto ha contratado recientemente al consultor Juan Cadaqués para estudiar el problema de los retrasos en el tráfico aéreo. Éste ha anotado el número de minutos de retraso de una muestra de vuelos en la siguiente tabla:$$
\begin{array}{llccccc}
\hline \begin{array}{l}
\text { Minutos } \\
\text { de retaso }
\end{array} & 0<10 & 10<20 & 20<30 & 30<40 & 40<50 & 50<60 \\
\hline \begin{array}{l}
\text { Número } \\
\text { de vueles }
\end{array} & 30 & 25 & 13 & 6 & 5 & 4 \\
\hline
\end{array}
$$
a) Estime el nümero medio de minutos de retraso.
b) Estime la varianza y la desviación típica muestrales.

Key Concepts

Grouped Frequency Distribution

A grouped frequency distribution is a way to organize data where individual data points are aggregated into defined intervals or groups. This method simplifies large data sets by summarizing the frequency (or count) of observations that fall within each specified class interval, allowing for easier computation of summary statistics and making patterns in the data more visible.

Class Midpoints

Class midpoints are the values that represent the center of each interval in a grouped frequency distribution. They are calculated by taking the average of the lower and upper bounds of an interval. These midpoints are used as representative values for each class, especially when estimating measures such as the mean and variance from grouped data.

Estimation of Sample Mean from Grouped Data

The sample mean for grouped data is estimated by multiplying each class midpoint by its corresponding frequency, summing these products, and then dividing by the total number of observations. This provides an approximate mean value because the exact data points within each interval are not known, and the midpoint is assumed to be a good representation of all values in that group.

Estimation of Sample Variance and Standard Deviation from Grouped Data

The estimation of sample variance from grouped data involves computing the squared deviation of each class midpoint from the estimated mean, weighting each squared deviation by the frequency of the respective class, and then dividing by the appropriate degrees of freedom (typically n-1 for an unbiased estimate). The sample standard deviation is subsequently obtained as the square root of the variance, providing a measure of dispersion that reflects how spread out the data are around the mean.

Transcript

00:01 So we have some data that is classes of data, and we're going to put down what the midpoint is.

00:06 And we have that those midpoints would be five minutes late, 15, 25, 35, 45, 45, and 55.

00:16 And then we have the frequency of those events, and that was 30, 25, 13, 6, 5, and 4.

00:30 And so let's find out.

00:31 I don't know if they told us how many that added up to.

00:34 It does not.

00:35 And that's the first piece of information we need.

00:38 So i have that data in my calculator, and i'm going to add up or sum up that second list so i can see how many flights that is.

00:47 And that is 83.

00:50 So now let's estimate what the mean is, the mean number of minutes late.

00:55 And so we're going to take our sum of all the numbers and divide by 80.

01:00 And we have five minutes late for 30 flights, 15 for 25, 25 for 13 of those, 35 for six of them, 45 for five of them, 45 for five of them, and finally 55 for four of them.

01:21 And let's calculate what that mean is.

01:27 And we find out that that mean comes out to be mean number of minutes late, comes out to be 18 .13 approximately.

01:36 So on the average, 18 .13 minutes late.

01:41 And then part b, we want to find the variance and the standard deviation.

01:47 So let's find the variance first.

01:48 We always find the variance first, regardless of whether we want to or not.

01:52 And we will be adding up a whole bunch of squared deviations and dividing by 82, one less than how many we have because this is a sample.

02:00 And so we're going to take our 5 minus the 18 .13 squared, and then how many of those do we have? we have 30 of those times 30...

Please give Ace some feedback