Class Exercise 13

Problem 1

In $2017$, in a sample of $245$ randomly selected apple trees in Saint-Joseph-du-Lac, it was found that $138$ had been infected with fire blight. Compute a $98\%$ confidence interval for the percentage of apple trees infected with fire blight in the apple orchads of Saint-Joseph-du-Lac in $2017$. Could you make a claim that more than half the apple trees are infected based on this sample?

Problem 2

Amelia's parents claim that it takes them on average $40$ minutes to put a meal on the table for dinner and therefore Amelia should wash the dishes and clean the kitchen after dinner, which according to the parents takes less than $30$ minutes. Amelia collected data for $81$ days which shows mean dinner preparation time of $37$ minutes with standard deviation of $8$ minutes and mean clean-up time of $28.5$ minutes with standard deviation of $10$ minutes.

For the dinner preparation time test $H_0:\mu=40$ against $H_1:\mu<40$ at $5\%$ level of significance. Are the parents correct in their claim it takes $40$ minutes to prepare dinner?
For the clean-up time test $H_0: \mu=30$ against $H_1: \mu<30$ at $5\%$ level of significance. Are the parents correct in their claim it takes less than 30 minutes to clean up after dinner?

Problem 3

A small brewery monitors the alcohol content of its beer by taking $6$ samples from every vat brewed. The label states that the alcohol content is $5\%$ and to avoid hefty fines the brewery discards all vats where a statistically significant deviation from $5\%$ is found. The following are the percent alcohol content data for the sample taken from tha last vat:$$5.3 \;\; 5.0\;\; 5.1\;\; 5.4\;\; 5.3\;\; 5.3$$

Calculate the mean and the standard deviation for the alcohol content of the sample.
Test the null hypothesis $H_0: \mu = 5$ against $H_1: \mu > 5$ at a $1\%$ level of significance. Report a $p$-value. Draw a conclusion in the context of the problem.

Problem 4

The table below shows the scores obtained by $9$ students in a Statistics class on Test $1$ and on Test $2$.$$ \begin{array}{l|ccccccccc} \text{Test }1 & 88 & 68 & 77 & 82 & 63 & 80 & 78 & 71 & 74\\ \hline \text{Test }2 & 73 & 77 & 67 & 74 & 74 & 64 & 71 & 71 & 72 \end{array} $$Test the null hypothesis $H_0: \mu_D = 0$ versus the alternative $H_1: \mu_D > 0$ at $5\%$ level of significance. Here, $\mu_D = \mu_1 - \mu_2$. Report a $p$-value. Draw a conclusion in the context of the problem.