sample problem\n• exam scores: 78, 70, 78, 89, 34, 95, 67, 88, 89, 78\n• mean, mode, median, standard…

sample problem\n• exam scores: 78, 70, 78, 89, 34, 95, 67, 88, 89, 78\n• mean, mode, median, standard deviation\n18.4. the exam scores of 30 students in a class follow\n64 73 62 70 83 41 49 85 96 30\n77 77 52 74 84 49 49 85 96 30\n79 64 91 66 75 78 41 48 82 75\n76 76 71 63 87 79 66 88 74 90\n(a) what is the lowest score in the class?\n(b) what is the mean score?\n(c) what is the mode?\n(d) compute the standard - deviation using equation 18.4.

sample problem\n• exam scores: 78, 70, 78, 89, 34, 95, 67, 88, 89, 78\n• mean, mode, median, standard deviation\n18.4. the exam scores of 30 students in a class follow\n64 73 62 70 83 41 49 85 96 30\n77 77 52 74 84 49 49 85 96 30\n79 64 91 66 75 78 41 48 82 75\n76 76 71 63 87 79 66 88 74 90\n(a) what is the lowest score in the class?\n(b) what is the mean score?\n(c) what is the mode?\n(d) compute the standard - deviation using equation 18.4.

Answer

Explanation:

Step1: Arrange scores in ascending order

$34,67,70,70,78,78,78,88,89,95$

Step2: Calculate the mean

Mean $\bar{x}=\frac{34 + 67+70+70+78+78+78+88+89+95}{10}=\frac{757}{10}=75.7$

Step3: Find the mode

The mode is the number that appears most frequently. Here, $78$ appears 3 times, so the mode is $78$.

Step4: Calculate the median

Since $n = 10$ (even), the median is the average of the $\frac{n}{2}$ - th and $(\frac{n}{2}+1)$ - th ordered values. $\frac{n}{2}=5$ and $\frac{n}{2}+1 = 6$. The 5 - th value is $78$ and the 6 - th value is $78$. So the median is $\frac{78 + 78}{2}=78$.

Step5: Calculate the standard deviation

First, calculate the squared differences from the mean: $(34 - 75.7)^2=(-41.7)^2 = 1738.89$ $(67 - 75.7)^2=(-8.7)^2=75.69$ $(70 - 75.7)^2=(-5.7)^2 = 32.49$ (twice) $(78 - 75.7)^2=(2.3)^2=5.29$ (thrice) $(88 - 75.7)^2=(12.3)^2 = 151.29$ $(89 - 75.7)^2=(13.3)^2=176.89$ $(95 - 75.7)^2=(19.3)^2 = 372.49$

The sum of squared differences $\sum_{i = 1}^{n}(x_i-\bar{x})^2=1738.89+75.69+2\times32.49+3\times5.29+151.29+176.89+372.49=2626.2$

The standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i - \bar{x})^2}{n-1}}=\sqrt{\frac{2626.2}{9}}\approx\sqrt{291.8}\approx17.08$

Answer:

Mean: $75.7$, Mode: $78$, Median: $78$, Standard Deviation: $\approx17.08$