QUESTION IMAGE
Question
- the histogram shows the daily high - temperature for grand rapids, michigan for each of the 30 days in june of a recent year. a. complete the table.
| temperature (degrees f) | frequency | relative frequency (%) | cumulative relative frequency (%) |
| 60 up to 65 | |||
| 65 up to 70 | |||
| 70 up to 75 | |||
| 75 up to 80 | |||
| 80 up to 85 | |||
| 85 up to 90 | |||
| 90 up to 95 |
b. what temperature interval contains the 40th percentile for this data?
- the medical college admission test (mcat) is a multiple - choice exam taken by students who are applying to medical school. the lowest possible score on the mcat is 472 and the highest possible score is 528. the table shows selected scores and their corresponding percentiles for recent year.
| score | 482 | 492 | 502 | 512 | 522 |
| percentile | 5 | 23 | 52 | 83 | 99 |
a. interpret the percentile for the mcat score of 512.
b. create a cumulative relative frequency graph using the information provided.
Step1: Determine frequencies from histogram
Count the number of data - points in each interval from the histogram. Let's assume the frequencies are as follows (since the histogram is not fully legible in text - form, we'll use a general approach). Suppose the frequencies for the intervals are \(f_1,f_2,f_3,f_4,f_5,f_6,f_7\) for the intervals \(60 - 65\), \(65 - 70\), \(70 - 75\), \(75 - 80\), \(80 - 85\), \(85 - 90\), \(90 - 95\) respectively.
Step2: Calculate relative frequencies
The relative frequency \(rf_i\) of each interval is calculated as \(rf_i=\frac{f_i}{n}\times100\%\), where \(n = 30\) (total number of days). For example, if \(f_1 = 2\), then \(rf_1=\frac{2}{30}\times100\%\approx6.67\%\).
Step3: Calculate cumulative relative frequencies
The cumulative relative frequency \(crf_i\) of the \(i\) - th interval is the sum of the relative frequencies of all intervals up to and including the \(i\) - th interval. For the first interval, \(crf_1=rf_1\). For the second interval, \(crf_2=rf_1 + rf_2\), and so on.
Step4: Find the 40th percentile
First, calculate \(0.4\times n=0.4\times30 = 12\). Then, find the interval where the cumulative frequency first exceeds 12.
Step5: Interpret MCAT percentile
The percentile of a score indicates the percentage of scores that are less than or equal to that score. For a MCAT score of 512 with a percentile of 83, it means that 83% of the students who took the MCAT scored 512 or lower.
Step6: Create cumulative relative - frequency graph for MCAT
Plot the scores on the x - axis and the cumulative relative frequencies (percentiles) on the y - axis. Mark the points \((482,5)\), \((492,23)\), \((502,52)\), \((512,83)\), \((522,99)\) and connect them with line - segments.
a.
| Temperature (degrees F) | Frequency | Relative frequency (%) | Cumulative relative frequency (%) |
|---|---|---|---|
| 65 up to 70 | Assume \(f_2\) | \(\frac{f_2}{30}\times100\) | \(\frac{f_1 + f_2}{30}\times100\) |
| 70 up to 75 | Assume \(f_3\) | \(\frac{f_3}{30}\times100\) | \(\frac{f_1 + f_2+f_3}{30}\times100\) |
| 75 up to 80 | Assume \(f_4\) | \(\frac{f_4}{30}\times100\) | \(\frac{f_1 + f_2+f_3 + f_4}{30}\times100\) |
| 80 up to 85 | Assume \(f_5\) | \(\frac{f_5}{30}\times100\) | \(\frac{f_1 + f_2+f_3 + f_4+f_5}{30}\times100\) |
| 85 up to 90 | Assume \(f_6\) | \(\frac{f_6}{30}\times100\) | \(\frac{f_1 + f_2+\cdots+f_6}{30}\times100\) |
| 90 up to 95 | Assume \(f_7\) | \(\frac{f_7}{30}\times100\) | \(\frac{f_1 + f_2+\cdots+f_7}{30}\times100\) |
b.
The temperature interval that contains the 40th percentile needs to be found by calculating cumulative frequencies from the histogram data. If, for example, after calculating cumulative frequencies, we find that the cumulative frequency first exceeds \(0.4\times30 = 12\) in the \(70 - 75\) interval, then the 40th percentile is in the \(70\) up to \(75\) temperature interval.
4.a.
A MCAT score of 512 is at the 83rd percentile. This means that 83% of the students who took the MCAT scored 512 or lower.
4.b.
The graph has x - axis labeled "MCAT Score" with values 482, 492, 502, 512, 522 and y - axis labeled "Percentile" with values 5, 23, 52, 83, 99. Plot the points \((482,5)\), \((492,23)\), \((502,52)\), \((512,83)\), \((522,99)\) and connect them with line - segments.
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Step1: Determine frequencies from histogram
Count the number of data - points in each interval from the histogram. Let's assume the frequencies are as follows (since the histogram is not fully legible in text - form, we'll use a general approach). Suppose the frequencies for the intervals are \(f_1,f_2,f_3,f_4,f_5,f_6,f_7\) for the intervals \(60 - 65\), \(65 - 70\), \(70 - 75\), \(75 - 80\), \(80 - 85\), \(85 - 90\), \(90 - 95\) respectively.
Step2: Calculate relative frequencies
The relative frequency \(rf_i\) of each interval is calculated as \(rf_i=\frac{f_i}{n}\times100\%\), where \(n = 30\) (total number of days). For example, if \(f_1 = 2\), then \(rf_1=\frac{2}{30}\times100\%\approx6.67\%\).
Step3: Calculate cumulative relative frequencies
The cumulative relative frequency \(crf_i\) of the \(i\) - th interval is the sum of the relative frequencies of all intervals up to and including the \(i\) - th interval. For the first interval, \(crf_1=rf_1\). For the second interval, \(crf_2=rf_1 + rf_2\), and so on.
Step4: Find the 40th percentile
First, calculate \(0.4\times n=0.4\times30 = 12\). Then, find the interval where the cumulative frequency first exceeds 12.
Step5: Interpret MCAT percentile
The percentile of a score indicates the percentage of scores that are less than or equal to that score. For a MCAT score of 512 with a percentile of 83, it means that 83% of the students who took the MCAT scored 512 or lower.
Step6: Create cumulative relative - frequency graph for MCAT
Plot the scores on the x - axis and the cumulative relative frequencies (percentiles) on the y - axis. Mark the points \((482,5)\), \((492,23)\), \((502,52)\), \((512,83)\), \((522,99)\) and connect them with line - segments.
a.
| Temperature (degrees F) | Frequency | Relative frequency (%) | Cumulative relative frequency (%) |
|---|---|---|---|
| 65 up to 70 | Assume \(f_2\) | \(\frac{f_2}{30}\times100\) | \(\frac{f_1 + f_2}{30}\times100\) |
| 70 up to 75 | Assume \(f_3\) | \(\frac{f_3}{30}\times100\) | \(\frac{f_1 + f_2+f_3}{30}\times100\) |
| 75 up to 80 | Assume \(f_4\) | \(\frac{f_4}{30}\times100\) | \(\frac{f_1 + f_2+f_3 + f_4}{30}\times100\) |
| 80 up to 85 | Assume \(f_5\) | \(\frac{f_5}{30}\times100\) | \(\frac{f_1 + f_2+f_3 + f_4+f_5}{30}\times100\) |
| 85 up to 90 | Assume \(f_6\) | \(\frac{f_6}{30}\times100\) | \(\frac{f_1 + f_2+\cdots+f_6}{30}\times100\) |
| 90 up to 95 | Assume \(f_7\) | \(\frac{f_7}{30}\times100\) | \(\frac{f_1 + f_2+\cdots+f_7}{30}\times100\) |
b.
The temperature interval that contains the 40th percentile needs to be found by calculating cumulative frequencies from the histogram data. If, for example, after calculating cumulative frequencies, we find that the cumulative frequency first exceeds \(0.4\times30 = 12\) in the \(70 - 75\) interval, then the 40th percentile is in the \(70\) up to \(75\) temperature interval.
4.a.
A MCAT score of 512 is at the 83rd percentile. This means that 83% of the students who took the MCAT scored 512 or lower.
4.b.
The graph has x - axis labeled "MCAT Score" with values 482, 492, 502, 512, 522 and y - axis labeled "Percentile" with values 5, 23, 52, 83, 99. Plot the points \((482,5)\), \((492,23)\), \((502,52)\), \((512,83)\), \((522,99)\) and connect them with line - segments.