ToddFennimore-hw1
Todd Fennimore
August 31, 2017
hw1 <- tbl_df(faithful)
hw1# A tibble: 272 x 2
eruptions waiting
* <dbl> <dbl>
1 3.600 79
2 1.800 54
3 3.333 74
4 2.283 62
5 4.533 85
6 2.883 55
7 4.700 88
8 3.600 85
9 1.950 51
10 4.350 85
# ... with 262 more rowsQuestion 1
==========
Histogram of waiting times between eruptions for Yellowstone Old Faithful Geyser
ggplot(hw1, aes(x=waiting)) + geom_histogram(bins = 30, col = "yellow", fill = "blue") + labs(title = "Waiting Times Between Eruptions for Old Faithful Geyser", x = "Waiting Times in Minutes")
Question 2
==========
The median is 76. The mean is 76, while the 80% trimmed mean is 71.5. The distribution has a negative or left skewed, since the mean is less than the median. The median would be the better statistic for average waiting time here given a unimodal normal distribution. However, one can see from the histogram that this distribution is bimodal, so it may be more accurate to say that one will usually either be waiting around 80 minutes between eruptions or sometimes around 55 minutes.
psych::describe(hw1 %>% select(waiting)) vars n mean sd median trimmed mad min max range skew
waiting 1 272 70.9 13.59 76 71.5 11.86 43 96 53 -0.41
kurtosis se
waiting -1.16 0.82Question 3
==========
The inter-quartile range is 24, while the standard deviation is 13.6. The standard deviation is most appropriate as a statistics for a normal distribution. Since this distribution is not normal, the IQR is the better measure of dispersion for the data.
hw1 %>%
summarize(IQR(waiting), quantile(waiting, 0.25), quantile(waiting, 0.75))# A tibble: 1 x 3
`IQR(waiting)` `quantile(waiting, 0.25)` `quantile(waiting, 0.75)`
<dbl> <dbl> <dbl>
1 24 58 82hw1 %>%
select(waiting) %>%
summarize_all(sd)# A tibble: 1 x 1
waiting
<dbl>
1 13.59497Question 4
==========
The distribution is bimodal. Upon inspection of the histogram (above), one can see that there are two local maxima, one around 55 and the other around 80.
Question 5
==========
The distribution is left or negative skewed, since the mean is less than the median. The summary statistics from the psych package in the table above confirms this, with a skew indicated of -0.41. When I use the skew1 statistic presented in class, I obtain a skew of -0.375. (Skew1 = (70.9 - 76)/13.6.) Values above 0.2 in absolute value generally indicate meaningful skew, according to Dr. Love’s class notes. By this statistic, there is, in fact, meaningful skew. The bimodal distribution is not symmetric.
==========
Question 6
==========
No, there are no outlier values for this distribution. From the IQR table above, we can calculate the upper fence as 83 + 1.5(24), or 119, and the lower fence as 58 + 1.5(24), or 22. Since our maximum value in the dataset is 96 and our minimum is 43, there are no outliers outside the upper or lower fences. One can also see from the box plot below that no values lie beyond the fences.
ggplot(hw1, aes(x = 1, y = waiting)) +
geom_boxplot(fill = "yellow") +
coord_flip() +
labs(title = "Boxplot of Waiting Times for Old Faithful",
y = "Waiting Times",
x = "") +
theme(axis.text.y = element_blank(),
axis.ticks.y = element_blank())
Question 7
==========
No, a model using a normal distribution would not be appropriate for summarizing this waiting time data. A normal distribution has a single mode, is symmetric, and is neither heavy-tailed nor light-tailed. This distribution of waiting data is bimodal and asymmetric. The Q-Q plot below confirms this non-normality, as the line is an S-shape curve characteristic of a skewed non-normal distribution.
ggplot(hw1, aes(sample = waiting)) +
geom_point(stat="qq") +
theme_bw() # eliminate the gray background
Question 8
==========
In general, one sees that as eruption duration increases, the waiting time between eruptions increases. This is a fairly strong positive correlation. In addition, one can see a general clustering of data between 0 and about 3 minutes and 3.5 and 5 minutes.
ggplot(hw1, aes(x=eruptions, y=waiting)) + geom_point() + labs(title = "Waiting Times vs. Eruption Duration in the Old Faithful Data")
##Question 9 ##==========
The correlation of waiting time with eruption duration is 0.9. This indicates a fairly strong positive correlation between waiting times and eruption duration, so that as one variable increases in value, so does the other.
cor(hw1$waiting, hw1$eruptions)[1] 0.9008112Question 10
===========
Yes, a linear model would be appropriate to use in attempting to predict the waiting time given the most recent eruption duration. One can see upon the inspection of the least squares regression line fitted to the scatterplot of the data that it is a fairly good model for the data. The linear equation, from values in the summary table below, is waiting time = 33.47 + 10.73(eruptions). The multiple R^2 is 81%, meaning that 81% of the variation inn the waiting time is explained by the eruption duration.
ggplot(hw1, aes(x = eruptions, y = waiting)) +
geom_point(size = 3) +
geom_smooth(method = "lm") +
labs(title = "Waiting Times by Eruption Duration",
x = "Eruption Duration", y = "Waiting Times")
summary(lm(waiting ~ eruptions, data = hw1))
Call:
lm(formula = waiting ~ eruptions, data = hw1)
Residuals:
Min 1Q Median 3Q Max
-12.0796 -4.4831 0.2122 3.9246 15.9719
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 33.4744 1.1549 28.98 <2e-16 ***
eruptions 10.7296 0.3148 34.09 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.914 on 270 degrees of freedom
Multiple R-squared: 0.8115, Adjusted R-squared: 0.8108
F-statistic: 1162 on 1 and 270 DF, p-value: < 2.2e-16Question 11
===========
This is another situation where inpection of the data in a scatterplot might be more meaningful than statistical measures without context. Looking at the scatterplot, we can interpret the data this way: if one observes an eruption lasting about 2 minutes, then one has probably waited 80 minutes from the last eruption to the current one; if one observes an eruption lasting about 4.5 minutes, then the waiting time could have been any time between about 50 and 90 minutes. The correlation is -0.64, meaning that there is an inverse relationship between waiting time and eruption duration: the longer the time between eruptions, the shorter the waiting time between eruptions. The R^2 is about 41%, meaning that about 41% of the variance in eruption duration is explained by waiting times between eruptions. This is a less powerful model than the other linear model, but still may apply. The linear equation for the least squares regression model here is: eruption duration = 7.31 - 0.05(waiting time).
hw1extra <- tbl_df(MASS::geyser)
hw1extra# A tibble: 299 x 2
waiting duration
* <dbl> <dbl>
1 80 4.016667
2 71 2.150000
3 57 4.000000
4 80 4.000000
5 75 4.000000
6 77 2.000000
7 60 4.383333
8 86 4.283333
9 77 2.033333
10 56 4.833333
# ... with 289 more rowsggplot(hw1extra, aes(x=duration, y=waiting)) + geom_point() + labs(title = "Waiting Times vs. Eruption Duration in the Geyser Data")
cor(hw1extra$duration, hw1extra$waiting)[1] -0.644623ggplot(hw1extra, aes(x = waiting, y = duration)) +
geom_point(size = 3) +
geom_smooth(method = "lm") +
labs(title = "Eruption Duration by Waiting Times",
x = "Waiting Times", y = "Eruption Duration")
summary(lm(duration ~ waiting, data = hw1extra))
Call:
lm(formula = duration ~ waiting, data = hw1extra)
Residuals:
Min 1Q Median 3Q Max
-2.21805 -0.72357 -0.01979 0.75071 2.11109
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.313144 0.269935 27.09 <2e-16 ***
waiting -0.053272 0.003666 -14.53 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.879 on 297 degrees of freedom
Multiple R-squared: 0.4155, Adjusted R-squared: 0.4136
F-statistic: 211.2 on 1 and 297 DF, p-value: < 2.2e-16```
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