---
title: "Homework assignment #9: Solutions"
author: "Milica Cudina"
date: "`r Sys.Date()`"
output: pdf_document
---
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```{r echo=FALSE, warning=FALSE, include=FALSE}
knitr::opts_chunk$set(
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  fig.align="center",
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)
```
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## Problem #1 (3 points)
  The mean area of the several thousand new apartments is advertised
  to be at least 1350 square feet.  A tenant group thinks that the
  apartments are smaller than advertised.  They hire an engineer to
  measure a sample of apartments, they should test
\begin{description}
\item[a.] $H_0: \mu=1350$ against $H_a: \mu > 1350.$
\item[b.] $H_0: \mu=1350$ against $H_a: \mu < 1350.$
\item[c.] $H_0: \mu<1350$ against $H_a: \mu = 1350.$
\item[d.] $H_0: \mu=1350$ against $H_a: \mu \not= 1350.$
\item[e.] $H_0: \mu<1350$ against $H_a: \mu > 1350.$
\end{description}


## Problem #2 (5 points)
In a hypothesis testing problem, $p-$value = 3\% means that \dots
\begin{description}
\item[a.] Null hypothesis has a 3\% chance to be wrong.
\item[b.] If the null hypothesis is true, the probability of observing
  as extreme or more extreme than what have been observed is 3\%.
\item[c.] Alternative hypothesis has a 3\% chance to be wrong.
\item[d.] If we repeat the procedure a lot times, 
approximately 3\% of the tests will be 
significant.
\item[e.] None of the above.
\end{description}


## Problem #2 (5 points)
  Freddie Threepwood conducts a hypothesis test. He calculates the
  observed value of the $z-$statistic to be $0.018$ (under the
  null). At the significance level of $0.05$, he should reject the
  null hypothesis. *True or false? Why?*


## Problem #3 (15 points)
  Suppose that the thumb sizes of the US males follow a normal
  distribution with an unknown mean $\mu$ and known standard deviation
  $\sigma=20$ on the GPI - scale (*Grey's Pollex Index - GPI -
    from $50$ to $280$*). The US Department of Thumbs and Toes (DTT)
  reports that the mean thumb size in the country is $\mu=150$.

  Being the chairman of the Faculty of Thumbs of the local university
  you see an excellent opportunity here and decide to conduct your own
  study of the size of the average American thumb. After collecting a
  SRS of 100 American thumbs you obtain the following sample average
  $\bar{x}=153$. 

\begin{itemize}
\item[{\bf i}] ($5$ pts) Construct a 95\%-confidence interval for the
  unknown parameter $\mu$ based on your study.
\item[{\bf ii.}] ($8$ pts)
  Assess the strength of evidence your study carries against the DTT findings. In other words: state the hypotheses and report the $p-$value.
\item[{\bf iii.}] ($2$ pts) You dream of achieving fame and fortune by
  being the first person ever to estimate the mean thumb size up to
  the margin of error equal to $\pm 0.1$. How large a sample size do
  you need for that? 
\end{itemize}


## Problem #4 (5 points)
*Source: "Probability and Statistics for Engineers and Scientists" by Walpole, Myers, Myers, and Ye.*

Assume that the compressive strength for a certain type of cement is normal with a known standard deviation of $120$ kilograms and an **unknown** mean $\mu.$ You test the hypotheses
\begin{equation}\nonumber
  \begin{split}
    H_0: \mu = 5000 \quad vs. \quad H_a: \mu < 5000. 
  \end{split}
\end{equation}
For a planned sample of size $50$, 
your colleague obtains the rejection region $RR=(-\infty, 4970]$. What is the significance level he used? 


## Problem #5 (7 points)
The *Cheesecake Manufacture and Dinery* claims that their famous cheesecake has at most $2000$ per slice. You suspect the contrary and plan a study. You model the calorie content per slice using the normal distribution with un unknown mean $\mu$ and with a **known** standard deviation of $300.$ 

You diligently study a random sample of $25$ slices of cheesecake. The sample average turns out to be $2100.$

What is the $p-$value corresponding to these data?


## Problem #6 (10 points)
The manufacturer of the *Slim Steakburger* brand claims that the
  mean fat content of this grade of steakburger is at most 18\%.

  The *Fat Fighters* consumer group, concerned about the mean fat
  content of this grade of steakburger submits to an independent
  laboratory a random sample of $12$ steakburgers for analysis. 

  Assuming the percentage fat content being normally distributed with
  a variance of $3$, they carry out an appropriate hypothesis test in
  order to advise the consumer group as to the validity of the
  manufacturer's claim.

The rejection region for the significance level of $0.05$ is 
$[18.8225, \infty)$. With the above significance level of
  $0.05$, find the power of the test at the alternative population mean of $20$.