U2 is an Irish band that has been popular for over three decades. The lengths of 80 of their songs were selected at random and are listed in the table below. Lengths of Randomly Selected U2 Songs. ?
(seconds) 382 222 214 338 310 559 334 225 177 189 278 300 295 254 331 194 225 294 245 292 262 219 384 336 312 281 308 285 227 258 331 418 195 338 255 225 43 331 179 230 215 252 272 228 281 274 229 178 173 319 267 280 66 186 341 204 253 317 303 337 308 240 252 349 347 94 144 187 272 208 279 276 264 270 286 205 350 444 294 260
a. Use 50-second intervals to make a frequency distribution table for the data.
c. Determine the mean, median, mode, range, and standard deviation for the song lengths. mean (s) median (s) mode (s) range (s) standard deviation (s)
d. Describe whether the data appears to be relatively normally distributed.
e. Assume the data is normally distributed and that the band’s entire collection of songs has a mean and standard deviation equal to those calculated above. What percentage of U2 songs are expected to be
i. over 180 seconds
ii. between 210 seconds and 300 seconds
f. Under what length of time are 90% of all U2 songs expected to be?
a. To create a frequency distribution table for the data, we need to group the song lengths into 50-second intervals. We will start by finding the minimum and maximum values in the data.
The minimum value in the data is 43 seconds, and the maximum value is 444 seconds.
We will use the following intervals:
0-49, 50-99, 100-149, 150-199, 200-249, 250-299, 300-349, 350-399, 400-449.
Now, we will count how many song lengths fall into each interval:
0-49: 1
50-99: 2
100-149: 1
150-199: 5
200-249: 12
250-299: 15
300-349: 13
350-399: 15
400-449: 6
So, the frequency distribution table is as follows:
Interval Frequency
0-49 1
50-99 2
100-149 1
150-199 5
200-249 12
250-299 15
300-349 13
350-399 15
400-449 6
c. To determine the mean, median, mode, range, and standard deviation for the song lengths, we can use statistical calculations.
Mean (average): The mean is calculated by summing up all the values and dividing by the total number of values. In this case, we have 80 song lengths.
Mean = (382 + 222 + 214 +...+ 294 + 260) / 80
Median: The median is the middle value when the data is arranged in ascending order. If there is an even number of data points, the median is the average of the two middle values.
First, we need to sort the data in ascending order:
43 66 94 144 173 177 178 179 186 187 189 194 195 204 205 208 214 215 219 222 225 225 225 227 228 229 230 240 245 252 252 253 254 255 258 260 262 264 267 270 272 272 274 276 279 280 281 281 285 286 292 294 295 300 303 308 308 310 312 317 319 331 331 334 336 337 338 338 341 347 349 350 382 384 444
Since we have 80 data points, the median will be the value at the 40th position, which is 274.
Mode: The mode is the value that appears most frequently in the data. In this case, there is no value that appears more than once, so there is no mode.
Range: The range is the difference between the maximum and minimum values. In this case, the range is 444 - 43 = 401 seconds.
Standard deviation: To calculate the standard deviation, we first need to find the variance. The variance is the average of the squared differences between each value and the mean.
Variance = [(382 - mean)^2 + (222 - mean)^2 + ... + (260 - mean)^2] / 80
Once we have the variance, we can calculate the standard deviation by taking the square root of the variance.
d. To determine whether the data appears to be relatively normally distributed, we can create a histogram or a normal probability plot of the data. The histogram will show the distribution of the song lengths, and if it resembles a bell-shaped curve, it suggests a normal distribution. The normal probability plot will plot the data against theoretical quantiles of a normal distribution. If the points on the plot align somewhat closely to a straight line, it indicates a normal distribution.
e. To calculate the percentages of U2 songs that are expected to fall into specific time intervals, assuming a normal distribution with the calculated mean and standard deviation, we can use the z-score formula.
i. To find the percentage of U2 songs expected to be over 180 seconds, we need to find the area under the normal distribution curve to the right of 180 seconds. We can calculate the z-score first using the formula:
z = (180 - mean) / standard deviation
Once we have the z-score, we can use a standard normal distribution table or a calculator to find the area to the right of the z-score. This will give us the percentage.
ii. To find the percentage of U2 songs expected to be between 210 and 300 seconds, we need to find the area under the normal distribution curve between those two values. We can calculate the z-scores for both values and then find the difference in the areas to get the percentage.
f. To determine the length of time under which 90% of all U2 songs are expected to be, we need to find the z-score that corresponds to a cumulative probability of 90% in the standard normal distribution. Then, we can use the formula:
length = (z-score * standard deviation) + mean
This will give us the length of time under which 90% of the songs are expected to be.
We do not do your homework for you. Although it might take more effort to do the work on your own, you will profit more from your effort. We will be happy to evaluate your work though.
However, I will give you a start.
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
I'll let you do the calculations.