ch4 q19: solve part d please The average playing time of music albums in a large collection is 37 minutes, and the standard deviation is 6 minutes.(a) What value is 1 standard deviation above the mean? 1 standard deviation below the mean? What values are 2 standard deviationsaway from the mean?(b) Without assuming anything about the distribution of times, at least what percentage of the times are between 25 and 49 minutes?(c) Without assuming anything about the distribution of times, what can be said about the percentage of times that are either less than19 minutes or greater than 55 minutes?(d) Assuming that the distribution of timesLess than 19 min or greater than 55 min? Less than 19 min?approximately normal, about what percentage of times are between 25 and 49 minutes?Step 1(a) What value is 1 standard deviation above the mean? 1 standard deviation below the mean? What values are 2 standard deviationsaway from the mean?To find the value that is some number of standard deviations above or below the mean, we will add or subtract some multiple of thestandard deviation, s, to the sample mean, x. The average playing time was given to be 37 minutes with a standard deviation of6 minutes, so we have x3737 and s = 6To find a value above the mean indicates we will use ladditionaddition. Finding a value below the mean indicates we will usesubtractionsubtraction. Finding values that are away from the mean includes values that are above and below the mean,indicating we will use addition and subtractionaddition and subtractionStep 2We know the mean is x = 37 and the standard deviation is s = 6. To find values above the mean indicates addition, values below the meanindicatesction, and both addition and subtraction will be usedfind values away frommean.Find the value that is 1 standard deviation above the mean.1 standard deviation above the mean = x + s= 4343Find the value that is 1 standard deviation below the mean.1 standard deviation below the mean = x - s= 37 - 6= 3131Find the values that are 2 standard deviations away from the mean.2 standard deviations above the mean = x + 25= 37 + 266492 standard deviations below mean = x - 2525 v25Step 3(b) Without assuming anything about the distribution of times, at least what percentage of the times are between 25 and49 minutes? (Round the answer to the nearest whole number.)When we assume nothing about the shape of the distribution, we will use Chebyshev's Rule to get a sense of the distribution of datavalues. Chebyshev's Rule will tell us the minimum percentageobservations that are within k standard deviations of the mean whenkz 1.We previously determined that 25 is two standard deviations below the mean and 49 is two standard deviations above the mean. Thus, wewill use k = 22 in Chebyshev's Rule to determine the minimum percentage of times that are between 25 and 49 minutes.110 -*2= 7575 %Thus, at least7575 % of times are between 25 and 49 minutes.Step 4(c) Without assuming anything about the distribution of times, what can be said about the percentagethan 19 minutes or greater than 55 minutes? (Round the answer to the nearest whole number.)times that are either lessAgain, we are not assuming anything about the shape of the distribution, so we will continue using Chebyshev's Rule. We will determinethe minimum percentage of times that are between 19 minutes and 55 minutes and then determine the maximum percentage of timesthat are less than 19 minutes or greater than 55 minutes. First, we need to know how many standard deviations 19 minutes and55 minutes are from the mean so we can determine the value for kWe previously determined that 43 minutes and 31 minutes are 1 standard deviation from the mean and that 49 minutes and 25 minutesare 2 standard deviations from the mean.2531374349?X- 3sX- 2sX-5X+5X+ 2sX+3sPerhaps9 minutes and 55 minutes are 3 standard deviations from the mean. Let's investigate.3 standard deviations above the meanx + 3s = 37 + 3655553 standard deviations below the meanx - 3s = 37 -= 1919We showed that 19 minutes and 55 minutes areare3 standard deviations from the mean.We showed that 19 minutes and 55 minutes areare 3 standard deviations from the mean.Step 5We determined that 19 minutes and 55 minutes are 3 standard deviations from the mean. Thus, we will use k = 3 3 in Chebyshev'sRule to determine the minimum percentagetimes that are between 19 and 55 minutes. (Round your final answer to the nearest wholenumber.)100(1 -% - 100 1 -(389However, we are asked to find the percentage of values that are less than 19 minutes and more than 55 minutes. These are values that arefound in the lower tail and in the upper tail of the distribution, not the middle. To find this number we will subtract the minimum percentage ofvalues that are between 3 standard deviations of the mean from 100%. (Round your answers to the nearest whole number.)100% - 8989 % = 11Thus, there will be no more than 1111 % of times less than 19 minutes or more than 55 minutes.(d) Assuming that the distribution of times is approximately normal, about what percentage of times are between 25 and 49 minutes? Lessthan 19 min or greater than 55 min? Less than 19 min? (Round the answers to two decimal places, if needed.)When we assume that the distribution of times is approximately normal, we will use the Empirical Rule to get a more precise sense of thedistribution of data values. The Empirical Rule gives us approximate percentages that are within k = 1, 2, or 3 standard deviations from theIf the distribution can be reasonably well approximated by a normal curve, then:• Approximately 68% of the observations are within 1 standard deviation of the mean.• Approximately 95% of the observations are within 2 standard deviations of the mean.• Approximately 99.7% of the observations are within 3 standard deviations of the mean.approximate percentages implied by the Empirical Rule34%34%2.35%13.5%13.5%2.35%-3 sd-2 sd-1 sdMean1 sd1sd1 sd2 sd3 sdIn contrast to Chebyshev's Rule, dividing the "within" percentages in half is permissible, because the normal curve is symmetric.Use the Empirical Rule to determine the percentage of times are between 25 and 49 minutes.We previously determinedapproximatelyminutes and 4minutesstaneviations of the mean. The Empirical Rule indicates% of times are between 25 minutes and 49 minutes.Use the Empirical Rule to determine the percentage of times that are less than 19 minutes or greater than 55 minutes.We previously determined that 19 minutes and 55 minutes are 3 standard deviations from the mean. The Empirical rule indicates thatapproximately% of times are between 19 minutes and 55 minutes.However, we were asked to determine the percentage of times that are less than 19 minutes or greater thanare found in the lower tail and in the upper tail of the distribution, not the middle. To find this number we will subtract the percentage of valuesminutes. These are values thatthat are between 3 standard deviations of the mean from 100%.100% -% =Approximately% of times are less than 19 minutesmore than 55 minutes.

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(d) Assuming that the distribution of times is approximately normal, about what percentage of times are between 25 and 49 minutes? Less than 19 min or greater than 55 min? Less than 19 min? (Round the answers to two decimal places, if needed.) When we assume that the distribution of times is approximately normal, we will use the Empirical Rule to get a more precise sense of the distribution of data values. The Empirical Rule gives us approximate percentages that are within k = 1, 2, or 3 standard deviations from the mean. If the distribution can be reasonably well approximated by a normal curve, then: • Approximately 68% of the observations are within 1 standard deviation of the mean. • Approximately 95% of the observations are within 2 standard deviations of the mean. • Approximately 99.7% of the observations are within 3 standard deviations of the mean. appraximate percentages implied by the Empirical Rule 34% 34% 2.35% 13.5% 13.5% 2.35% -3 sd -2 sd Mean 1sd 2 sd 3 sd 1 sd 1 sd In contrast to Chebyshev's Rule, dividing the "within" percentages in half is permissible, because the normal curve is symmetric. Use the Empirical Rule to determine the percentage of times are between 25 and 49 minutes. We previously determined that 25 minutes and 49 minutes are 2 standard deviations of the mean. The Empirical Rule indicates that approximately 7% of times are between 25 minutes and 49 minutes. Use the Empirical Rule to determine the percentage of times that are less than 19 minutes or greater than 55 minutes. We previously detemined that 19 minutes and 55 minutes are 3 standard deviations from the mean. The Empirical rule indicates that approximately % of times are between 19 minutes and 55 minutes. However, we were asked to determine the percentage of times that are less than 19 minutes or greater than 55 minutes. These are values that are found in the lower tail and in the upper tail of the distribution, not the middle. To find this number we will subtract the percentage of values that are between 3 standard deviations of the mean from 100%. 100% - % = % Approximately % of times are less than 19 minutes or more than 55 minutes.