Write R code to answer the quiz questions on Learn using the dataset below.

Investigators studied physical characteristics and ability in 13 football punters. Each volunteer punted a football ten times. The investigators recorded the average distance for the ten punts, in feet. They also recorded the average hang time (time the ball is in the air before the receiver catches it) for the ten punts, in seconds. In addition, the investigators recorded five measures of strength and flexibility for each punter: right leg strength (pounds), left leg strength (pounds), right hamstring muscle flexibility (degrees), left hamstring muscle flexibility (degrees), and overall leg strength (foot-pounds). From the study “The relationship between selected physical performance variables and football punting ability” by the Department of Health, Physical Education and Recreation at the Virginia Polytechnic Institute and State University, 1983.

```
Variable Description
------------- --------------------------------
Distance Distance travelled in feet
Hang Time in air in seconds
R_Strength Right leg strength in pounds
L_Strength Left leg strength in pounds
R_Flexibility Right leg flexibility in degrees
L_Flexibility Left leg flexibility in degrees
O_Strength Overall leg strength in pounds
```

Data File: `ADA2_WS_01_punting.csv`

The Relationship Between Selected Physical Performance Variables and Football Punting Ability. Department of Health, Physical Education and Recreation, Virginia Polytechnic Institute and State University, 1983.

- Read the data set into R.

```
library(tidyverse)
# First, download the data to your computer,
# save in the same folder as this Rmd file.
# read the data
dat_punt <- read_csv("ADA2_WS_01_punting.csv", skip = 1)
str(dat_punt)
```

```
Classes 'spec_tbl_df', 'tbl_df', 'tbl' and 'data.frame': 13 obs. of 7 variables:
$ Distance : num 162 144 148 164 192 ...
$ Hang : num 4.75 4.07 4.04 4.18 4.35 4.16 4.43 3.2 3.02 3.64 ...
$ R_Strength : num 170 140 180 160 170 150 170 110 120 130 ...
$ L_Strength : num 170 130 170 160 150 150 180 110 110 120 ...
$ R_Flexibility: num 106 92 93 103 104 101 108 86 90 85 ...
$ L_Flexibility: num 106 93 78 93 93 87 106 92 86 80 ...
$ O_Strength : num 241 195 153 197 267 ...
- attr(*, "spec")=
.. cols(
.. Distance = col_double(),
.. Hang = col_double(),
.. R_Strength = col_double(),
.. L_Strength = col_double(),
.. R_Flexibility = col_double(),
.. L_Flexibility = col_double(),
.. O_Strength = col_double()
.. )
```

- (7 p) Generate summaries
`summary()`

and frequency tables`table()`

for each variable. Answer questions 1–7.

```
Distance Hang R_Strength
Min. :104.9 Min. :3.020 Min. :110.0
1st Qu.:140.2 1st Qu.:3.640 1st Qu.:130.0
Median :150.2 Median :4.040 Median :150.0
Mean :148.2 Mean :3.921 Mean :147.7
3rd Qu.:163.5 3rd Qu.:4.180 3rd Qu.:170.0
Max. :192.0 Max. :4.750 Max. :180.0
L_Strength R_Flexibility L_Flexibility
Min. :110.0 Min. : 85.00 Min. : 78.00
1st Qu.:130.0 1st Qu.: 90.00 1st Qu.: 86.00
Median :150.0 Median : 93.00 Median : 93.00
Mean :143.8 Mean : 95.69 Mean : 91.23
3rd Qu.:160.0 3rd Qu.:103.00 3rd Qu.: 94.00
Max. :180.0 Max. :108.00 Max. :106.00
O_Strength
Min. :130.2
1st Qu.:153.9
Median :197.1
Mean :196.2
3rd Qu.:240.6
Max. :266.6
```

```
$Distance
104.93 105.67 117.59 140.25 144 147.5 150.17 162
1 1 1 1 1 1 1 1
162.5 163.5 165.17 171.75 192
1 1 1 1 1
$Hang
3.02 3.2 3.6 3.64 3.68 3.85 4.04 4.07 4.16 4.18 4.35
1 1 1 1 1 1 1 1 1 1 1
4.43 4.75
1 1
$R_Strength
110 120 130 140 150 160 170 180
1 2 1 2 1 2 3 1
$L_Strength
110 120 130 140 150 160 170 180
2 1 2 1 3 1 2 1
$R_Flexibility
85 86 89 90 92 93 95 101 103 104 106 108
1 1 1 1 2 1 1 1 1 1 1 1
$L_Flexibility
78 80 83 86 87 92 93 94 95 106
1 1 1 1 1 1 3 1 1 2
$O_Strength
130.24 132.68 152.99 153.92 154.64 195.49 197.09 205.88
1 1 1 1 1 1 1 1
219.25 240.57 260.56 266.56
1 2 1 1
```

Note that you can do even better than reading the numbers from above to answer the specific quiz questions. Instead, you can (not required) write code that returns the specific values you want. For example:

The minimum distance is 104.93 ft.

- What was the lowest recorded punting distance among the 13 participants?
- What was the highest recorded hang time among the 13 participants?
- Is the range of values for R_Strength the same or different than the range of values for L_Strength?
- What percentage of the sample has a L_Strength of 110 pounds?
- Is the range of values for R_Flexibility the same or different than the range of values for L_Flexibility?
- What percentage of the sample has a L_Flexibility of 106 degrees?
- What is the most common value for O_Strength (i.e., what is the modal value)?

- (2 p) Plot \(y=\)
`Distance`

and \(x=\)`Hang`

and interpret the plot in terms of linearity and strength of correlation.

(2 p) Calculate the Pearson correlation between

`Distance`

and`Hang`

(read the help for performing the hypothesis test). Answer questions 8–9.What is the correlation between Distance and Hang?

The corresponding p-value for the correlation between Distance and Hang is ____.

(2 p) Create a new categorical (factor) variable,

`O_StrengthFac`

, from the quantitative variable overall leg strength (`O_Strength`

) to indicate high leg strength: code less than 200 as 0 (low leg strength) and at least 200 as 1 (high leg strength).

Plot \(y=\)`Distance`

and \(x=\)`O_StrengthFac`

and interpret the comparison of distance by strength group.

```
# plot distance by strength group
library(ggplot2)
# p <- ggplot(dat_punt, aes(x = , y = ))
# ...
# print(p)
```

(2 p) Use a two-sample \(t\)-test (assume equal variance) to test whether \(H_0: \mu_{\textrm{low}} = \mu_{\textrm{high}}\), that the population means for distance are equal for the two overall leg strength groups you created. Answer questions 10–11.

Is distance significantly associated with overall strength (categorical) at an alpha = 0.05 level?

What is the mean distance in feet for the low and high strength groups, respectively?

(2 p) Plot \(y=\)

`Distance`

and \(x=\)`R_Flexibility`

and interpret the relationship.

(2 p) Regress \(y=\)

`Distance`

on \(x=\)`R_Flexibility`

. Answer questions 12–13.What is the expected increase in distance for each degree increase in flexibility?

Is distance significantly associated with flexibility at an alpha = 0.05 level?

(2 p) Create a new variable which is the mean of the right leg and left leg flexibility variables,

`O_Flexibility`

. Generate a frequency distribution for this new variable. Answer questions 14–15.What is the median value for your new variable that is the mean of the right and left leg flexibility?

What percentage of the sample has a mean flexibility no more than 86 degrees?

(4 p) Upload your error-free program (html output as PDF file) showing your work and your plots for additional points.

For these questions above:

- (2 p) plot and interpretation.

- (2 p) plot and interpretation.

- (2 p) plot and interpretation.

- (4 p) code and output appear correct, no errors.