Question

calculate the mean for the frequency distribution of the test scores below.
test scores | frequency
40 - 49 | 5
50 - 59 | 9
60 - 69 | 10
70 - 79 | 6
80 - 89 | 3
90 - 99 | 8
round your answer to 2 decimal places.
mean =
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question 9
2 pts 1 details
score on last try: 2 of 2 pts. see details for more.
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match each distribution below with its best descriptor. (roughly symmetric, left skewed, right skewed, bimodal, uniform.) then identify whether the mean, median, or both would be the best measure of spread.
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Explanation:

Step1: Find midpoints of each class

For \(40 - 49\), midpoint \(x_1=\frac{40 + 49}{2}=44.5\)
For \(50 - 59\), midpoint \(x_2=\frac{50 + 59}{2}=54.5\)
For \(60 - 69\), midpoint \(x_3=\frac{60 + 69}{2}=64.5\)
For \(70 - 79\), midpoint \(x_4=\frac{70 + 79}{2}=74.5\)
For \(80 - 89\), midpoint \(x_5=\frac{80 + 89}{2}=84.5\)
For \(90 - 99\), midpoint \(x_6=\frac{90 + 99}{2}=94.5\)

Step2: Calculate \(f_i\times x_i\) for each class

\(f_1\times x_1 = 5\times44.5 = 222.5\)
\(f_2\times x_2 = 9\times54.5 = 490.5\)
\(f_3\times x_3 = 10\times64.5 = 645\)
\(f_4\times x_4 = 6\times74.5 = 447\)
\(f_5\times x_5 = 3\times84.5 = 253.5\)
\(f_6\times x_6 = 8\times94.5 = 756\)

Step3: Find total frequency (\(N\)) and total \(f_i\times x_i\)

\(N=5 + 9 + 10 + 6 + 3 + 8=41\)
Total \(f_i\times x_i=222.5+490.5 + 645+447+253.5+756 = 2814.5\)

Step4: Calculate mean

Mean \(\bar{x}=\frac{\sum f_i x_i}{N}=\frac{2814.5}{41}\approx68.6463\)

Step5: Round to two decimal places

Rounded mean \(= 68.65\) (Wait, wait, let's recalculate \(\frac{2814.5}{41}\): \(41\times68 = 2788\), \(2814.5 - 2788 = 26.5\), \(26.5\div41\approx0.646\), so \(68 + 0.646\approx68.65\)? Wait no, wait \(41\times68.6463\approx2814.5\). Wait let's do exact division: \(2814.5\div41 = 68.646341\cdots\), so rounded to two decimal places is \(68.65\)? Wait no, 68.6463, the third decimal is 6, so we round up the second decimal: 68.65? Wait no, 68.6463, first decimal 6, second 4, third 6. So 68.65? Wait wait, no: 68.6463, when rounding to two decimal places, look at the third digit which is 6, so we add 1 to the second digit: 4 + 1 = 5. So 68.65. Wait but let's check the calculations again.

Wait total \(f_i x_i\): 222.5 + 490.5 = 713; 713 + 645 = 1358; 1358 + 447 = 1805; 1805 + 253.5 = 2058.5; 2058.5 + 756 = 2814.5. Total frequency N = 5 + 9 = 14; 14 + 10 = 24; 24 + 6 = 30; 30 + 3 = 33; 33 + 8 = 41. So mean is 2814.5 / 41. Let's compute 4168 = 2788, 2814.5 - 2788 = 26.5. 26.5 / 41 = 0.6463. So 68.6463, which is 68.65 when rounded to two decimal places. Wait but wait, maybe I made a mistake in midpoints? Let's check midpoints again: 40-49: (40+49)/2=44.5, correct. 50-59: 54.5, correct. 60-69:64.5, correct. 70-79:74.5, correct. 80-89:84.5, correct. 90-99:94.5, correct. Frequencies: 5,9,10,6,3,8. Correct. So total f_i x_i: 544.5=222.5, 954.5=490.5, 1064.5=645, 674.5=447, 384.5=253.5, 8*94.5=756. Sum: 222.5+490.5=713; 713+645=1358; 1358+447=1805; 1805+253.5=2058.5; 2058.5+756=2814.5. Correct. N=41. So 2814.5/41=68.6463, so rounded to two decimal places is 68.65? Wait no, 68.6463, the second decimal is 4, third is 6, so we round up the second decimal: 4 becomes 5, so 68.65. Yes.

Answer:

\(68.65\)