Statistics 528: Homework 2 Solutions

Transcription

1 Statistics 28: Homework 2 Solutions.4 There are several gaps in the data, as can be seen from the histogram. Minitab Result: Min Q Med Q3 Max Manual Result: Min Q Med Q3 Max We cannot find out the overall shape or distribution from the five-number summary. From histogram, we can see there may be 3 modes, also a gap between $6,000 & $9,000, which couldn t be seen from summaries..4 (a) Stem-and-Leaf Display: Fees Stem-and-leaf of Fees N = 0 Leaf Unit = (28) $ is the most common value. (b) IQR=4, and Q3=22, so anything over 28 is a possible outlier. There are several values over $.

2 .0 Without the outlier, the mean is 37.9 (compared to before 4.06). The median is 37 (compared to 38.). The mean changed by about 3., while the median changed by only. as a result of dropping the outlier..8 (a) Back-to-back stemplot: Leaf unit: 0 minutes Symmetric, bell-shaped distributions without outliers are better summarized using x and s. (b) Descriptive Statistics: Minutes by Gender: (Including all data) Variable Gender N Mean Median TrMean StDev Minutes F M Variable Gender SE Mean Minimum Maximum Q Q3 Minutes F M The question asks if it appears to contain a high outlier. So we only need the upper limit. Female: upper limit=q3+.*iqr=+.*(-)=2 Male: upper limit=q3+.*iqr=7.+.*(7.-)=3.7 In the female group, 3 is flagged as suspicious by.*iqr criterion. In the male group, 0 is marginally within.*iqr. Descriptive Statistics: Minutes by Gender (excluding the highest outlier in each group) Variable Gender N Mean Median TrMean StDev Minutes F M Variable Gender SE Mean Minimum Maximum Q Q3 Minutes F M For female students, removing the outlier 3 resulted in the change of the mean from 6.2 to8.4 and standard deviation from 6. to Likewise, for male students

3 the mean decreases by 6.3 from 7.2 to 0.9 and the standard deviation decreases by 7.3 from 74.2 to 66.9 when we remove the outlier..72 (a) a=0, b=/0.62=.63; 6mph=04.8km/h Xnew=a+bX, when X=0, Xnew=0, it means a=0. Xnew=bX When X=0.62, Xnew=, it means b=/0.62=.63. Xnew=.63X When X=6, Xnew=.63*6=04.8km/h (b) a=0, b=746; hp=04,4watts Xnew=a+bX, when X=0, Xnew=0, it means a=0. Xnew=bX When X=, Xnew=746, it means b=746. Xnew=746X When X=, Xnew=*746=04,4watts.78 (a) The curve forms a box that is unit wide and unit high. Therefore, the area is *= (b) The shaded area is 0.8 wide and unit high. Therefore, the area is For Eleanor, z=(6-00)/00=.8. For Gerald, z=(27-8)/6=.. Eleanor's score is higher..88 (a) P(Z<2.8)=0.78 (b) P(Z>2.8)=- P(Z<2.8)=-0.78= (c) P(Z>-.66)=P(Z<.66)=0.9 by the symmetry of the standard normal distribution (d) P(-.66<Z<2.8)=P(Z<2.8)-P(Z<-.66)=

4 (a) Let X denote the yearly return, can be approximated by the N(3, 7). By the criterion, approximately 9% of years lie within the range of 2 standard deviation from the mean. Therefore, from 3%-2*7%=-2% to 3+2*7%=47% (b) The percent of years when the yearly return is below 0 is P(X<0)=P((X-3)/7<(0-3)/7)=P(Z<-0.76)=0.22 (c) The percent of years when the yearly return is above 2% is P(X>2)=P((X- 3)/7>(2-3)/7)=P(Z>0.6)= On the right end, the data are higher than predicted for a normal curve. Note that the Minitab uses the data values for the x-axis and the normal percentiles for the y-axis. Therefore, when the data on the right end fall below the straight line, it indicates a heavier tail than normal distribution or skewness to the right. (Note: this is different from the text book since normal quantile plots in the text use the data values for the y-axis.) Likewise, when data on the left end fall above the straight line, it indicataes skewness to the left. Normal Probability Plot for Fees - 9% CI Mean.9 StDev 7.62 AD*

5 . 4 3 GPA SEX

6 0 0 SATM SEX From the plots, we see that the medians of GPA are fairly similar for both groups. The median of men's SATM is higher than that of women's. There seems no difference in the spread of two distributions for men and women. Among the four distributions below, the distribution of male students'satm has a normal quantile plot with most of the points falling approximately on the straight line in the middle, and within the band. Normal Probability Plot for GPA of Men - 9% CI Mean StDev AD*

7 Normal Probability Plot for GPA of Women - 9% CI Mean 2.68 StDev AD*

8 Normal Probability Plot for SATM of Men - 9% CI Mean StDev AD* Normal Probability Plot for SATM of Women - 9% CI Mean 6.02 StDev AD*