Probstat/week13 practice 1

This is part of probstat

Reviews

1. We toss a fair coin 100 times. Let random variable $X$ be the number of heads.

• Find the minimum value $T$ such that $P\{|X-50|>T\}\leq 0.05$. You may use computers to compute $T$ exactly.
• Find the minimum value $T$ such that $P\{X>50+T\}\leq 0.05$. You may use computers to compute $T$ exactly.
• Find the minimum value $T$ such that $P\{|X-50|>T\}\leq 0.01$. You may use computers to compute $T$ exactly.
• Find the minimum value $T$ such that $P\{X>50+T\}\leq 0.01$. You may use computers to compute $T$ exactly.

Hint: In this question, you can calculate the probability $P\{X=i\}$ for $i=0,\ldots ,100$ because $X$ is a binomial random variable.

2. What is the variance of $X$ from question 1?

3. Let's approximate $X$ from question 1 with the normal distribution.

• Find the minimum value $T$ such that $P\{|X-50|>T\}\leq 0.05$. Use the standard normal distribution table to answer this. Compare with the answer from question 1.
• Find the minimum value $T$ such that $P\{X>50+T\}\leq 0.05$. Use the standard normal distribution table to answer this. Compare with the answer from question 1.
• Find the minimum value $T$ such that $P\{|X-50|>T\}\leq 0.01$. Use the standard normal distribution table to answer this. Compare with the answer from question 1.
• Find the minimum value $T$ such that $P\{X>50+T\}\leq 0.01$. Use the standard normal distribution table to answer this. Compare with the answer from question 1.

Tests concerning the mean of a population: known $\sigma$

1. An ISP claims to provide an Internet connection with the speed of 10Mbps. You know that the speed is normally distributed with s.d. $\sigma =1.5$. You plan to obtain a sample of size 10 of the Internet connection speed, and compute the sample mean ${\bar {X}}$. Design 3 statistical tests with levels of significances $\alpha$ = 0.1, 0.05, and 0.01 to show if we should believe the claim from the ISP. Define explicitly what your null hypothesis $H_{0}$ is.

Hint: your tests should specify the conditions to reject $H_{0}$ and should depend on ${\bar {X}}$.

2. Consider the tests you design from question 1. What is the probability that the tests accept $H_{0}$ when the actual speed is 9.5Mbps?

Tests concerning the mean of a population: unknown $\sigma$

1. An ISP claims to provide an Internet connection with the speed of 10Mbps. You know that the speed is normally distributed but you do not know $\sigma$. You plan to obtain a sample of size 10 of the Internet connection speed, and compute the sample mean ${\bar {X}}$ and the sample standard deviation $S$. Design 3 statistical tests with levels of significances $\alpha$ = 0.1, 0.05, and 0.01 to show if we should believe the claim from the ISP. Define explicitly what your null hypothesis $H_{0}$ is.

Hint: your tests should specify the conditions to reject $H_{0}$ and should depend on ${\bar {X}}$ and $S$.

2. Consider the tests you design from question 1. What is the probability that the tests accept $H_{0}$ when the actual speed is 9.5Mbps?

Tests concerning the equality means of normal population: known $\sigma$

We have two populations: students who attend special track-and-field training and students who do not attend the training. Let's call the average time for a 100m-run of the first group of students $\mu _{x}$ and of the second group of students $\mu _{y}$. Suppose that we know that the run time are normally distributed with standard deviation $\sigma =2$ seconds.

Suppose that we randomly choose $n$ students from the first group and find the sample mean ${\bar {X}}$ of their times for 100m runs. Also, we randomly choose $m$ students from the second group and find the sample mean ${\bar {Y}}$ of times of 100m runs.

We would like to know if the training has any effects on student run times.

1. What is the distribution of ${\bar {X}}$?

2. What is the distribution of ${\bar {Y}}$?

3. What is the distribution of ${\bar {X}}-{\bar {Y}}$?

4. Describe statistical tests of significant levels $\alpha$ = 0.1, 0.05, and 0.01.