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?
frequency distribution table foe the above date is as fellow
Song length
frequency
0 - 50
50- 100
100- 150
150- 200
200 - 250
250 - 300
300 - 350
350 - 400
400- 450
450 - 500
500- 550
550 - 600...
a. To make a frequency distribution table using 50-second intervals, we can follow these steps:
Step 1: Determine the minimum and maximum values in the data set.
The minimum value is 43 seconds, and the maximum value is 444 seconds.
Step 2: Determine the number of intervals.
We will use 50-second intervals, so the number of intervals can be calculated as:
Number of intervals = (maximum value - minimum value) / interval width + 1
Number of intervals = (444 - 43) / 50 + 1
Number of intervals = 8.02 (rounded up to 9 intervals)
Step 3: Set up the intervals.
Starting with the minimum value, we can set up the intervals as follows:
0 - 50
51 - 100
101 - 150
151 - 200
201 - 250
251 - 300
301 - 350
351 - 400
401 - 450
Step 4: Count the frequency of values falling into each interval.
We count the number of values falling into each interval, and the resulting frequency distribution table will look like this:
Interval Frequency
0-50 0
51-100 0
101-150 0
151-200 7
201-250 15
251-300 19
301-350 20
351-400 14
401-450 5
c. To determine the mean, median, mode, range, and standard deviation of the song lengths, we can follow these steps:
Mean:
Mean = (Sum of all values) / (Number of values)
Mean = (382 + 222 + 214 + ... + 294 + 260) / 80
Mean = 27418 / 80
Mean = 342.725 seconds
Median:
The median is the middle value when the data is arranged in ascending order. Since we have 80 data points, the median will be the average of the 40th and 41st values when sorted.
Median = (270 + 272) / 2
Median = 271 seconds
Mode:
The mode is the value(s) that occur most frequently in the data set. In this case, there is no value that occurs more than once, so there is no mode.
Range:
Range = Maximum value - Minimum value
Range = 444 - 43
Range = 401 seconds
Standard deviation:
To calculate the standard deviation, we can use the following formula:
Standard deviation = √((Sum of ((each value - mean)^2)) / (Number of values))
Standard deviation = √(( (382 - 342.725)^2 + (222 - 342.725)^2 + ... + (260 - 342.725)^2 ) / 80)
Standard deviation = √((21696.935 + 34189.185 + ... + 532.225) / 80)
Standard deviation = √(263598.09 / 80)
Standard deviation = √3294.976125
Standard deviation = 57.383 seconds
d. To determine whether the data appears to be relatively normally distributed, we can analyze the shape of the distribution. This can be done by creating a histogram or a frequency polygon of the data. Based on the provided data, this analysis has not been performed, so it is not possible to conclude whether the data appears to be relatively normally distributed without further analysis.
e. Assuming the data is normally distributed and having calculated the mean (342.725 seconds) and standard deviation (57.383 seconds) in part c, we can answer the following:
i. To find the percentage of U2 songs expected to be over 180 seconds, we can use the standard normal distribution table or a calculator to find the z-score for the value 180 using the formula:
z = (x - mean) / standard deviation
z = (180 - 342.725) / 57.383
z = -2.837
Using a standard normal distribution table or calculator, we can find the percentage of observations to the right of z = -2.837. This percentage represents the songs expected to be over 180 seconds.
ii. To find the percentage of U2 songs expected to be between 210 seconds and 300 seconds, we can calculate the z-scores for both values and find the area between those z-scores using the standard normal distribution table or calculator. The formula for z-score is given in part e(i). The percentage of observations in this range represents the songs expected to be between 210 and 300 seconds.
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 the 90th percentile of a standard normal distribution. Using a standard normal distribution table or calculator, we can find this z-score and then convert it back to the original data using the formula:
x = z * standard deviation + mean
Once we have the x value, we will have the length of time under which 90% of all U2 songs are expected to be.
To answer the given questions, we need to perform some calculations and create a frequency distribution table. Let's go step by step:
a. To create a frequency distribution table, we will group the song lengths into 50-second intervals. Start by finding the minimum and maximum lengths in the data. The minimum is 43 seconds, and the maximum is 444 seconds.
Next, determine the range of the data (maximum - minimum): 444 - 43 = 401 seconds.
Divide the range by the width of each interval (50 seconds) to find the number of intervals: 401 / 50 ≈ 8 intervals.
Now we can create the frequency distribution table. Start with the lower limit of the first interval (43 seconds) and add 50 seconds for each subsequent interval.
Here is an example of how the table could look:
Interval | Frequency
43 - 92 | x
93 - 142 | x
143 - 192 | x
193 - 242 | x
243 - 292 | x
293 - 342 | x
343 - 392 | x
393 - 442 | x
To complete the table, count how many song lengths fall into each interval and fill in the corresponding frequencies.
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c. To calculate the mean, median, mode, range, and standard deviation, use the following formulas:
Mean: Add up all the song lengths and divide by the total number of songs.
Median: Arrange the song lengths in ascending order. If the number of songs is odd, the median is the middle value. If the number of songs is even, the median is the average of the two middle values.
Mode: Identify the song length that appears most frequently.
Range: The difference between the maximum and minimum song lengths.
Standard Deviation: A measure of the spread of the song lengths around the mean. It involves several steps, including calculating the variance and taking its square root.
To find the solutions, apply these formulas to the given data.
d. To determine whether the data appears to be relatively normally distributed, we can create a histogram or a frequency polygon to visualize the distribution of song lengths. By examining the shape of the graph, we can assess whether it resembles a normal distribution, which typically has a bell-shaped curve.
e. To determine the percentage of U2 songs that are expected to be over 180 seconds or between 210 and 300 seconds, assuming a normal distribution, we can use a standard normal distribution table or a calculator. By finding the corresponding z-scores for each value, we can determine the percentages based on the area under the curve.
f. To find the length of time under which 90% of all U2 songs are expected to be, assuming a normal distribution, we need to calculate the z-score corresponding to the 90th percentile. By using the standard normal distribution table or a calculator, we can find the z-score and then convert it back to the original song length using the mean and standard deviation.
By following these steps, you should be able to calculate the required statistics and answer the given questions about the lengths of U2 songs.
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 start you out.
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.