Exercise 1

Let’s start by looking at the distribution of height_m, the typical height in meters for a Pokemon species, using a visualization and summary statistics.

ggplot(data = pokemon, aes(x = height_m)) +
  geom_histogram(binwidth = 0.25, fill = "steelblue", color = "black") + 
  labs(x = "Height (in meters)", 
       y = "Distributon of Pokemon heights")

pokemon %>%
  summarise(mean_height = mean(height_m), 
            sd_height = sd(height_m), 
            n_pokemon = n())

In the previous lecture (and in the review questions), we were given the mean, μ, and standard deviation, σ, of the population. That is unrealistic in practice (if we knew μ and σ, we wouldn’t need to do statistical inference!).

Today we will use our sample data and the Central Limit Theorem to draw conclusions about the μ, the mean height in the population of Pokemon.

• What is the point estimate for μ, i.e., the “best guess” for the mean height of all Pokemon?

• What is the point estimate for σ, i.e., the “best guess” for the standard deviation of the distribution of Pokemon heights?

Exercise 2

Before moving forward, let’s check the conditions required to apply the Central Limit Theorem. Are the following conditions met:

• Independence?
• Sample size/distribution?

Exercise 3

By the Central Limit Theorem,

ˉx∼N(μ,σ/√n)⇒Z=ˉx−μσ/√n

where Z is a standardized score such that Z∼N(0,1).

• Describe the distribution of Z in words.

In practice, we can’t calculate the standardized score Z, so instead we will use the standardized score T when conducting inference for a population mean…

Z=ˉx−μσ/√n⇒T=ˉx−μ0s/√n

• How do Z and T differ?

• What is the estimated standard error s/√n for the Pokemon data?

# add code

T is a new standardized score that follows a t distribution with n−1 degrees of freedom. It is written as tn−1. We will use the tn−1 distribution to help us conduct hypothesis tests and construct confidence intervals.

Exercise 4

The mean height of humans is about 1.65 meters. We would like to test whether the mean height of Pokemon is less than the mean height of humans.

• State the null and alternative hypotheses in words and statistical notation.

• Calculate the T test statistic.

T=ˉx−μ0s/√n

where μ0 is the null hypothesized value.

# add code

• What is the distribution of the test statistic, T?

• Now let’s calculate the p-value. Fill in the code below to use the pt() function to calculate the p-value. For x input the value of the test statistic, and for df input the degrees of freedom.

#pt(x = ____, df = ____)

• State with the p-value means.

• State the conclusion in the context of the data using a significance level of α=0.05.

Exercise 5

We would like to construct a 90% confidence interval for the mean height of Pokemon species. The equation general equation for a confidence interval is

estimate±crit∗×SE

Specifically, the confidence interval for the mean is

ˉx±t∗n−1×s√n

The second part of the equation, t∗n−1×s√n is called the margin of error.

We already know ˉx and s/√n, so let’s talk about tn−1. This value is determined based on the confidence level, C. It is the point on the t distribution with n−1 degrees of freedom, such that the area between −t∗ and t∗ is C.

• What is the critical value t∗ for our 90% confidence interval of the mean Pokemon height?

## add code

• Now calculate the 90% confidence interval for the mean Pokemon height.

# add code

• Interpret the interval in the context of the data.

CLT-based calculations in infer

pokemon %>%
  t_test(response = height_m, 
         alternative = "less", 
         mu = 1.65, 
         conf_int = FALSE)

pokemon %>%
  t_test(response = height_m, 
         conf_int = TRUE, 
         conf_level = 0.9) %>%
  select(lower_ci, upper_ci)