Instructions

The goal of this analysis is to use inference based on the Central Limit Theorem to analyze the mean synergy rating of burritos.

1. We’ll start by examining the distribution of synergy, a rating indicating how well all the ingredients in the burrito come together.

   • Visualize the distribution of synergy using a histogram with binwidth of 0.5.

   • Calculate the following summary statistics: the mean synergy, standard deviation of synergy, and sample size size. Save the summary statistics as summary_stats. Then display summary_stat.

2. The goal of this analysis is to use CLT-based inference to understand the true mean synergy rating of all burritos. The idea is that if CLT holds, we can assume the distribution of the sample mean is normal and thus easily generate a normal null distribution to test hypotheses.

Based on the data, what is your “best guess” for the mean synergy rating of burritos?

3. Is the synergy in burritos generally good? To answer this question, we will conduct a hypothesis test to evaluate if the mean synergy is greater than 3.

Before conducting inference, we need to check the conditions to make sure the Central Limit Theorem can be applied in this analysis. For each condition, indicate whether it is satisfied and provide a brief explanation supporting your response.

 - Independence? 
 - Sample size / distribution? 

4. State the null and alternative hypotheses to evaluate the question posed in the previous exercise. Write the hypotheses in words and in statistical notation.

5. Let \(\bar{x}\) be the mean synergy score in a sample of 330 randomly selected burritos. Given the Central Limit Theorem and the hypotheses from the previous exercise

• What is the mean of the sampling distribution of \(\bar{x}\)? In other words, what is the mean under the null?
• What is the standard error of the sampling distribution of \(\bar{x}\)?

6. Next, use R as a “calculator” to calculate the test statistic, \(T\). Recall the formula for the test statistic:

\[T = \frac{\bar{x}- \mu_{0}}{s/\sqrt{n}}\] where \(\bar{x}\) is the sample mean, \(\mu_0\) is the mean under the null, \(s\) is the sample s.d. and \(n\) is the sample size.

• Explain what this value means in the context of this analysis. Refer to sliders here from the preparation for last class if necessary.

• What is the distribution of the test statistic, \(T\)? Be specific. Hint: It is ___ distributed with ___ degrees of ____.

7. Now let’s calculate the p-value and draw a conclusion.

• Use the pt() function to calculate the p-value.
• Explain what this value means in the context of this analysis.
• State your conclusion in the context of the data using a significance level of \(\alpha = 0.01\).

8. Now let’s calculate a 90% confidence interval for the mean synergy rating of all burritos. The confidence interval for a population mean is

\[\bar{x} \pm t^*_{n-1} \times \frac{s}{\sqrt{n}}\]

We already know \(\bar{x}\) and \(\frac{s}{\sqrt{n}}\), so let’s focus on the calculating \(t^*_{n-1}\). We will use the qt() function to calculate the critical value \(t^*_{n-1}\).

Here is an example: If we want to calculate a 95% confidence interval for the mean, we will use qt(0.975, n-1), where 0.975 is the cumulative probability at the upper bound of the 95% confidence interval (recall we used this value to find the upper bound when calculating bootstrap confidence intervals), and (n-1) are the degrees of freedom.

• Calculate the critical value, \(t^*_{n-1}\), of the 90% confidence interval for the mean synergy rating of all burritos.

• Use R as a “calculator” to calculate the 90% confidence interval.

• Interpret the interval in the context of the data.

9. In the previous exercises, we conducted a hypothesis test and calculated a confidence interval step-by-step. We can also use infer for the calculations in CLT-based inference using the t_test() function.

• Fill in the code below to produce the calculations for the hypothesis test you conducted in Exercises 4 - 7.

⊕The results should be the same as the calculations you did in exercises in the previous exercises.

burritos %>%
  t_test(response = _____, 
         alternative = "______", 
         mu = ______, 
         conf_int = FALSE)

• Next, fill in the code below to produce the 90% confidence interval for the mean synergy rating.

⊕The results should be the same as the calculations from Exercise 8.

burritos %>%
  t_test(response = _____, 
         conf_int = TRUE, 
         conf_level = _____) %>%
  select(lower_ci, upper_ci)

10. Now let’s compare inference simulation-based inference versus inference using the Central Limit Theorem.

• What is one way conducting hypothesis tests is the same in both approaches? What is one way conducting hypothesis tests differs between the two approaches?
• What is one way calculating and interpreting confidence intervals is the same in both approaches? What is one way calculating and interpreting confidence intervals differs between the two approaches?