In 2004, the state of North Carolina released a large data set containing information on births recorded in this state. This data set is useful to researchers studying the relation between habits and practices of expectant mothers and the birth of their children. We will work with a random sample of observations from this data set.

Load the `nc`

data set into our workspace.

`library(ggplot2)`

`## Warning: package 'ggplot2' was built under R version 3.4.4`

`load("more/nc.RData")`

We have observations on 13 different variables, some categorical and some numerical. The meaning of each variable is as follows.

variable | description |
---|---|

`fage` |
father’s age in years. |

`mage` |
mother’s age in years. |

`mature` |
maturity status of mother. |

`weeks` |
length of pregnancy in weeks. |

`premie` |
whether the birth was classified as premature (premie) or full-term. |

`visits` |
number of hospital visits during pregnancy. |

`marital` |
whether mother is `married` or `not married` at birth. |

`gained` |
weight gained by mother during pregnancy in pounds. |

`weight` |
weight of the baby at birth in pounds. |

`lowbirthweight` |
whether baby was classified as low birthweight (`low` ) or not (`not low` ). |

`gender` |
gender of the baby, `female` or `male` . |

`habit` |
status of the mother as a `nonsmoker` or a `smoker` . |

`whitemom` |
whether mom is `white` or `not white` . |

- What are the cases in this data set? How many cases are there in our sample?

**The cases in this data set are births in the state of North Carolina. There are 1000 cases in the sample.**

As a first step in the analysis, we should consider summaries of the data. This can be done using the `summary`

command:

`summary(nc)`

```
## fage mage mature weeks
## Min. :14.00 Min. :13 mature mom :133 Min. :20.00
## 1st Qu.:25.00 1st Qu.:22 younger mom:867 1st Qu.:37.00
## Median :30.00 Median :27 Median :39.00
## Mean :30.26 Mean :27 Mean :38.33
## 3rd Qu.:35.00 3rd Qu.:32 3rd Qu.:40.00
## Max. :55.00 Max. :50 Max. :45.00
## NA's :171 NA's :2
## premie visits marital gained
## full term:846 Min. : 0.0 married :386 Min. : 0.00
## premie :152 1st Qu.:10.0 not married:613 1st Qu.:20.00
## NA's : 2 Median :12.0 NA's : 1 Median :30.00
## Mean :12.1 Mean :30.33
## 3rd Qu.:15.0 3rd Qu.:38.00
## Max. :30.0 Max. :85.00
## NA's :9 NA's :27
## weight lowbirthweight gender habit
## Min. : 1.000 low :111 female:503 nonsmoker:873
## 1st Qu.: 6.380 not low:889 male :497 smoker :126
## Median : 7.310 NA's : 1
## Mean : 7.101
## 3rd Qu.: 8.060
## Max. :11.750
##
## whitemom
## not white:284
## white :714
## NA's : 2
##
##
##
##
```

As you review the variable summaries, consider which variables are categorical and which are numerical. For numerical variables, are there outliers? If you aren’t sure or want to take a closer look at the data, make a graph.

Consider the possible relationship between a mother’s smoking habit and the weight of her baby. Plotting the data is a useful first step because it helps us quickly visualize trends, identify strong associations, and develop research questions.

- Make a side-by-side boxplot of
`habit`

and`weight`

. What does the plot highlight about the relationship between these two variables?

The plot shows that the median weight of babies with mothers who are smokers is lower than the median weight of babies with mothers who are not smokers.

`ggplot(data = nc, aes(x=habit, y=weight)) + geom_boxplot()`

The box plots show how the medians of the two distributions compare, but we can also compare the means of the distributions using the following function to split the `weight`

variable into the `habit`

groups, then take the mean of each using the `mean`

function.

`by(nc$weight, nc$habit, mean)`

```
## nc$habit: nonsmoker
## [1] 7.144273
## --------------------------------------------------------
## nc$habit: smoker
## [1] 6.82873
```

There is an observed difference, but is this difference statistically significant? In order to answer this question we will conduct a hypothesis test.

- Check if the conditions necessary for inference are satisfied. Note that you will need to obtain sample sizes to check the conditions. You can compute the group size using the same
`by`

command above but replacing`mean`

with`length`

.

*Random*: The babies included in this dataset are selected randomly.

*Normal*: The sample size of each group is reasonably large (>30).

`by(nc$weight, nc$habit, length)`

```
## nc$habit: nonsmoker
## [1] 873
## --------------------------------------------------------
## nc$habit: smoker
## [1] 126
```

*Independent*: The birth statistics of each baby are independent (individual babies do not affect each other). Also, the sample size is less than 10% of the population.

- Write the hypotheses for testing if the average weights of babies born to smoking and non-smoking mothers are different.

\(H_0\): The mean weight of babies born to smoking mothers is equal to the mean weight of babies born to non-smoking mothers.

\(H_A\): The mean weight of babies born to smoking mothers is not equal to the mean weight of babies born to non-smoking mothers.

Next, we introduce a new function, `inference`

, that we will use for conducting hypothesis tests and constructing confidence intervals.

```
DATA606::inference(y = nc$weight, x = nc$habit, est = "mean", type = "ht", null = 0,
alternative = "twosided", method = "theoretical")
```

```
## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_nonsmoker = 873, mean_nonsmoker = 7.1443, sd_nonsmoker = 1.5187
## n_smoker = 126, mean_smoker = 6.8287, sd_smoker = 1.3862
```

```
## Observed difference between means (nonsmoker-smoker) = 0.3155
##
## H0: mu_nonsmoker - mu_smoker = 0
## HA: mu_nonsmoker - mu_smoker != 0
## Standard error = 0.134
## Test statistic: Z = 2.359
## p-value = 0.0184
```

Let’s pause for a moment to go through the arguments of this custom function. The first argument is `y`

, which is the response variable that we are interested in: `nc$weight`

. The second argument is the explanatory variable, `x`

, which is the variable that splits the data into two groups, smokers and non-smokers: `nc$habit`

. The third argument, `est`

, is the parameter we’re interested in: `"mean"`

(other options are `"median"`

, or `"proportion"`

.) Next we decide on the `type`

of inference we want: a hypothesis test (`"ht"`

) or a confidence interval (`"ci"`

). When performing a hypothesis test, we also need to supply the `null`

value, which in this case is `0`

, since the null hypothesis sets the two population means equal to each other. The `alternative`

hypothesis can be `"less"`

, `"greater"`

, or `"twosided"`

. Lastly, the `method`

of inference can be `"theoretical"`

or `"simulation"`

based.

- Change the
`type`

argument to`"ci"`

to construct and record a confidence interval for the difference between the weights of babies born to smoking and non-smoking mothers.

```
DATA606::inference(y = nc$weight, x = nc$habit, est = "mean", type = "ci", null = 0,
alternative = "twosided", method = "theoretical")
```

```
## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_nonsmoker = 873, mean_nonsmoker = 7.1443, sd_nonsmoker = 1.5187
## n_smoker = 126, mean_smoker = 6.8287, sd_smoker = 1.3862
```

```
## Observed difference between means (nonsmoker-smoker) = 0.3155
##
## Standard error = 0.1338
## 95 % Confidence interval = ( 0.0534 , 0.5777 )
```

By default the function reports an interval for (\(\mu_{nonsmoker} - \mu_{smoker}\)). We can easily change this order by using the `order`

argument:

```
inference(y = nc$weight, x = nc$habit, est = "mean", type = "ci", null = 0,
alternative = "twosided", method = "theoretical",
order = c("smoker","nonsmoker"))
```

```
## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_smoker = 126, mean_smoker = 6.8287, sd_smoker = 1.3862
## n_nonsmoker = 873, mean_nonsmoker = 7.1443, sd_nonsmoker = 1.5187
```

```
## Observed difference between means (smoker-nonsmoker) = -0.3155
##
## Standard error = 0.1338
## 95 % Confidence interval = ( -0.5777 , -0.0534 )
```

- Calculate a 95% confidence interval for the average length of pregnancies (
`weeks`

) and interpret it in context. Note that since you’re doing inference on a single population parameter, there is no explanatory variable, so you can omit the`x`

variable from the function.

The 95% confidence interval for the average length of pregnancies is (38.1528 , 38.5165). This means that 95% of random samples will have an average pregnancy length that falls in that that interval.

```
inference(y = nc$weeks, est = "mean", type = "ci", null = 0,
alternative = "twosided", method = "theoretical")
```

```
## Single mean
## Summary statistics:
```