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The box-and-whisker plot below represents some data set. What percentage of the data values are less than or equal to 110 ?
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Based on the box-and-whisker plot, it can be seen that the lower whisker extends down to 85 and the box starts at 95. This means that the minimum value of the data set is 85 and the first quartile (Q1) is 95.
To determine the percentage of data values that are less than or equal to 110, we need to find the third quartile (Q3) of the data set. From the box-and-whisker plot, we can see that the upper whisker extends up to 150 and the box ends at 125. This means that the maximum value of the data set is 150 and the third quartile (Q3) is 125.
Therefore, the interquartile range (IQR) of the data set is Q3-Q1 = 125-95 = 30.
To find the percentage of data values that are less than or equal to 110, we need to find the z-score of 110 using the formula:
z = (x - mean) / standard deviation
Since we do not know the mean and standard deviation of the data set, we can use the formula for the z-score of a value relative to the quartiles:
z = (x - Q1) / IQR
if x is between Q1 and Q3
z = (x - Q3) / IQR
if x is above Q3
z = (110 - 95) / 30 = 0.50
According to the standard normal distribution table, the area to the left of z = 0.50 is 0.6915 or 69.15%. Therefore, approximately 69.15% of the data values are less than or equal to 110.
Step-by-step solution:
  1. Find the minimum value of the data set: 85
  2. Find the first quartile (Q1) of the data set: 95
  3. Find the third quartile (Q3) of the data set: 125
  4. Find the interquartile range (IQR) of the data set: Q3-Q1 = 125-95 = 30
  5. Calculate the z-score of 110 using the quartiles: z = (x - Q1) / IQR = (110 - 95) / 30 = 0.50
  6. Use the standard normal distribution table to find the area to the left of z = 0.50, which is approximately 0.6915 or 69.15%
  7. Therefore, approximately 69.15% of the data values are less than or equal to 110.
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