```from IPython.core.display import HTML, Markdown, display

import numpy.random as npr
import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
import scipy.stats as stats
import statsmodels.formula.api as smf
import pingouin as pg
import math

import ipywidgets as widgets

# Enable plots inside the Jupyter Notebook
%matplotlib inline
```

# Differences between means¶

Authored by Todd Gureckis and Brenden Lake with input from Matt Crump.

## Exercise 1: Bootstrapping the t-distribution¶

```ts=[]
for _ in range(10000): # repeat 10000 times
r_sample = np.random.normal(0,1,size=XXX) #set size size according to instruction
sem = np.std(r_sample,ddof=1)/np.sqrt(len(r_sample))
t_stat = np.mean(r_sample)/sem
ts.append(t_stat)

ts2=[]
for _ in range(10000):
r_sample = np.random.normal(0,1,size=XXX) #set size according to instructions
sem = np.std(r_sample,ddof=1)/np.sqrt(len(r_sample))
t_stat = np.mean(r_sample)/sem
ts2.append(t_stat)

sns.displot(ts)
plt.xlim([-10,10])
sns.displot(ts2)
plt.xlim([-10,10])
```
```---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-1-dd6554b5d4fb> in <module>
1 ts=[]
2 for _ in range(10000): # repeat 10000 times
----> 3     r_sample = np.random.normal(0,1,size=XXX) #set size size according to instruction
4     sem = np.std(r_sample,ddof=1)/np.sqrt(len(r_sample))
5     t_stat = np.mean(r_sample)/sem

NameError: name 'np' is not defined
```

## Exercise 2: Relationship between p and t values¶

```@widgets.interact(dof=widgets.IntSlider(min=1, max=53, step=1, value=10), alpha=widgets.FloatSlider(min=0,max=0.5, step=0.01, value=0.2))
def plot_t_onsided(dof, alpha):
fix, ax = plt.subplots(1,1,figsize=(10,6))

x=np.linspace(-3.5,3.5,100)
y=stats.t.pdf(x,df=dof)
t_crit=stats.t.ppf(1.0-alpha, df=dof)
print(t_crit)
ax.plot(x,y)
ax.set_ylabel("probability")
ax.set_xlabel("value of t statistic")
ax.set_title("One Sided Test")
ax.fill_between(x,y,where=x>t_crit,interpolate=True,facecolor='lightblue',alpha=0.2,hatch='/',edgecolor='b')
ax.set_xticks([0, t_crit])
#ax.set_yticklabels([])

sns.despine(top=True, right=True, left=True)

plt.show()
```

## Exercise 3: Computing a one sample t-test by hand¶

```# Your answer here
scores=np.array([.5,.56,.76,.8,.9])  # here is your data
# compute the "effect" (i.e., difference between the mean of the values and the null hypothesis)
# compute the error (i.e., the standard error of the mean), pay attention to whether you are dividing by n-1 or n
# compute the t-value
# use stats.t.cdf() to compute the area in the tail of the correct t-distribution for a one sided test.
```

## Exercise 4: Using `pingouin` to do a on sample `ttest()`¶

```scores=np.array([.5,.56,.76,.8,.9])
print(np.mean(scores))
pg.ttest(x=scores, y=0.25, alternative='greater')
```

## Exercise 5: Paired t-test example¶

### STUDY DESCRIPTION¶

Parents often sing to their children and, even as infants, children listen to and look at their parents while they are singing. Research by Mehr, Song, and Spelke (2016) sought to explore the psychological function that music has for parents and infants, by examining the hypothesis that particular melodies convey important social information to infants. Specifically, melodies convey information about social affiliation.

The authors argue that melodies are shared within social groups. Whereas children growing up in one culture may be exposed to certain songs as infants (e.g., “Rock-a-bye Baby”), children growing up in other cultures (or even other groups within a culture) may be exposed to different songs. Thus, when a novel person (someone who the infant has never seen before) sings a familiar song, it may signal to the infant that this new person is a member of their social group.

To test this hypothesis, the researchers recruited 32 infants and their parents to complete an experiment. During their first visit to the lab, the parents were taught a new lullaby (one that neither they nor their infants had heard before). The experimenters asked the parents to sing the new lullaby to their child every day for the next 1-2 weeks.

Following this 1-2 week exposure period, the parents and their infant returned to the lab to complete the experimental portion of the study. Infants were first shown a screen with side-by-side videos of two unfamiliar people, each of whom were silently smiling and looking at the infant.The researchers recorded the looking behavior (or gaze) of the infants during this ‘baseline’ phase. Next, one by one, the two unfamiliar people on the screen sang either the lullaby that the parents learned or a different lullaby (that had the same lyrics and rhythm, but a different melody). Finally, the infants saw the same silent video used at baseline, and the researchers again recorded the looking behavior of the infants during this ‘test’ phase.For more details on the experiment’s methods, please refer to Mehr et al. (2016) Experiment 1.

```# get the baby data frame
# filter to only have the data from experiment 1
experiment_one_df = baby_df[baby_df['exp1']==1]
```

### Baseline phase: Conduct a one sample t-test¶

You first want to show that infants’ looking behavior did not differ from chance during the baseline trial. The baseline trial was 16 seconds long. During the baseline, infants watched a video of two unfamiliar people, one of the left and one on the right. There was no sound during the baseline. Both of the actors in the video smiled directly at the infant.

The important question was to determine whether the infant looked more or less to either person. If they showed no preference, the infant should look at both people about 50% of the time. How could we determine whether the infant looked at both people about 50% of the time?

The `experiment_one_df` data frame has a column called `Baseline_Proportion_Gaze_to_Singer`. All of these values show how the proportion of time that the infant looked to the person who would later sing the familiar song to them. If the average of these proportion is .5 across the infants, then we would have some evidence that the infants were not biased at the beginning of the experiment. However, if the infants on average had a bias toward the singer, then the average proportion of the looking time should be different than .5.

Using a one-sample t-test, we can test the hypothesis that our sample mean for the `Baseline_Proportion_Gaze_to_Singer` was not different from .5.

```# here is how to get the column
experiment_one_df['Baseline_Proportion_Gaze_to_Singer']

```

Remember how the experiment went. Infants watched silent video recordings of two women (Baseline). Then each person sung a song, one was familiar to the infant (their parents sung the song to them many times), and one was unfamiliar (singing phase). After the singing phase, the infants watched the silent video of the two singers again (test phase). The critical question was whether the infants would look more to the person who sung the familiar song compared to the person who sun the unfamiliar song, which is recorded as `Test_Proportion_Gaze_to_Singer`. If the infants did this, they should look more than 50% of the time to the singer who sang the familiar song. We have the data, we can do another one sample t-test to find out.

```# here is how to get the column
experiment_one_df['Test_Proportion_Gaze_to_Singer']