## $**`$Task $2\mathrm{.}7`**\{-\}$

What is the conclusion of your hypothesis test at the $1\%$ significance level? Is the observed proportion $`$xbar$`**$significantly different$**$ from $20\%?$ What assumptions do we need about our data in order to make our hypothesis test valid? Your answer should have three things:

$1.$ At most one sentence stating the conclusion of your hypothesis test.

$2.$ Three reasons for your conclusion. At most three sentences.

$3.$ At most two sentences explaining what assumptions we used during our hypothesis test.

$**`$Answer to Task $2\mathrm{.}7$:$`**$

$1.$

$2.$

$3.$

# Question $3\{-\}$

## $**`$Task $3\mathrm{.}1`**\{-\}$

What is the the smallest prediction interval at which the data is consistent with the hypothesis? What percentage does this prediction interval correspond to$?$ What do you notice about this interval? Write your answer after the $**`$Answer to Task $3\mathrm{.}1`**$ prompt provided below.

$```\{$r$\}$

### Code for Task $3\mathrm{.}1.$ Write your code here

$```$

$**`$Answer to Task $3\mathrm{.}1$:$`**$

## $**`$Task $3\mathrm{.}2`**\{-\}$

Suppose we only had access to $n$ days from Autumn $2023,$ where $n $<92$$$.$ If we assume that $`$xbar$`$ would remain the same as in Task $2\mathrm{.}1$ regardless of the value of $n$$,$ what is the smallest value $n$ such that we would reject the hypothesis with a $**95\%**$ significance level? Hint: you may need to do go through some trial and error. Your answer should have two things:

$1.$ R code which prints appropriate output;

$2.$ one or two explanatory sentences $($strictly no more than two$).$

$```\{$r$\}$

### Code for Task $3\mathrm{.}2.$ Write your code here

$```$

$**`$Answer to Task $3\mathrm{.}2$:$`**$