Probability & Statistics Honors Unit Test #2

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 1 Each month, the percentage of non-working streetlights that were repaired is recorded. A sample of these measurements is shown above. Describe the distribution. (5)

The distribution of percent of repaired streetlights is skew left, centered near 90%, and has an approximate range between 15% (80-95%) and 25% (75-100%). There are no unusual features.

 2 Ariel’s height is listed as being in the 67th percentile. Explain what this means. (3)

Ariel’s height is equal to or greater than 67% of all other heights.

 350 560 106 1133 142 744 9300 500 99 144 960 353 44 1540 99 111 3269 372 877 33

 3 The New York Department of Environmental Conservation records the number of trout stocked into each lake in New York. A sample of those data is shown above. Determine if there are any outliers in the data. (5)

I’ll use Tukey’s Rule of Thumb—anything lower than ${Q}_{1}-1.5\left(IQR\right)$ or greater than ${Q}_{3}+1.5\left(IQR\right)$ will be considered an outlier. For these data, ${Q}_{1}=108.5$ and ${Q}_{3}=918.5$, so $IQR=810$. The lower limit for outliers is ${Q}_{1}-1.5\left(IQR\right)=-1106.5$ and the upper limit is ${Q}_{3}+1.5\left(IQR\right)=2133.5$. There are no data below the lower limit, but there are two data above the upper limit (3269 and 9300), so there are outliers in this data set.

The California Department of Transportation routinely collects weather data in the city of Pasadena. The data shown above are the daily high temperature (°F) for a random sample of days.

 83 102 99 71 83 78 64 73 93 84 63 75 79 93 97 94 71 81 60 61

 4 Find the mean of the data. (2)

80.2°F

 5 Find the median of the data. (2)

80°F

 6 Find the range of the data. (2)

42°F

 7 Find the standard deviation of the data. (2)

13.0771°F

 8 Find ${Q}_{1}$ for these data. (2)

71°F

 9 Find ${Q}_{3}$ for these data. (2)

93°F

 10 Find the interquartile range of these data. (2)

22°F

 11 Determine which measures of center and spread are best for these data. Justify your answer. (4)

I’ll look at the graph of the data first.

This looks skew right to me, which would indicate that the median and interquartile range are the better measures for center and spread.

The City of Los Angeles obtains water from two sources—fresh water (from rainfall and streams) and recycled water (water that has been used in the city once already). The data below are the amount of recycled water used in Los Angeles (in acre-feet) for a random selection of months.

 1104 143 231 976.8 443 835 874 1159 575 980.2 1282 249 1111 1295 380 1061 815 1125 833 578

 12 Construct a boxplot of these data. (5)

Ecologists in Brazil measured the diameter (cm) of a sample of trees from the Tapajos National Forest in two different years. Samples from the data collected each year are given below.

1999 Measurements

 36.5 16.5 40.97 13.5 66.8 21.3 24.4 17.8 50.6 23

2001 Measurements

 81.7 71.1 35.7 51.9 133.7 10.5 18.2 12.6 43.3 13.5

 13 Compare the distributions of these two samples. (5)

Let’s look at the graphs first.

The center of the 1999 distribution is lower than that of the 2001 distribution. The spread of tree diameters in 1999 is smaller than that of the 2001 distribution. Both distributions are skew right. Neither distribution has any unusual features.

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