Answer to Question #232914 in Statistics and Probability for eliakim ndume

Question

Suppose the measures to ensure that all manufactured motor vehicles meet requirements with regards to carbon monoxide emission indicate that the standard deviation per unit volume of exhaust fumes must not be in excess of 0.03 percent carbon monoxide emission particles per unit volume. A random sample of ten newly manufactured motor vehicles are tested and the following carbon monoxide emissions are recorded: 0.23, 0.27, 0.31, 0.25, 0.30, 0.29, 0.32, 0.26, 0.24, 0.22

A). Calculate the point estimate of the standard deviation of the carbon monoxide emissions.

B). What is the 99% confidence interval estimate for the population variance of the carbon monoxide emissions?

C). Provide an interpretation for the answer in b).

D). Suppose the level of confidence is reduced, what will happen to the answer in b)? Motivate. 

E). Use the critical value approach and test a hypothesis at a 5% level of significance to determine whether the requirements are met? Show all hypothesis test steps. 

Answer 1

Part a

$s^2=\frac{1}{n-1}[\sum x_i^2- \frac{1}{n} \sum(x_i)^2]\\ s^2=\frac{1}{10-1}[0.7345- \frac{1}{10} (2.69)^2]\\ s^2=0.0348$

Part b

$CI= [\frac{(n-1)s^2}{x^2}_{\alpha /2}, \frac{(n-1)s^2}{x^2}_{1-\alpha /2}]\\ CI= [\frac{(10-1)0.0012}{23.5894}, \frac{(10-1)0.0012}{1.7349}]\\ CI= [0.0005,0.0063]\\$

Part c

We are 99% confident that the population variance lies between 0.0005 and 0.0063

Part d

When the confidence level is reduced the lower limit will be increased and the upper limit will be reduced.

Part e

We are given that the standard deviation of the emission should not be in excess of 0.03.

$\sigma \ngtr0.03 \implies \sigma ^2= 0.0009$

Statistical Hypothesis – a conjecture about a population parameter. This conjecture may or may not be true.

There are two types of statistical hypotheses:

Null Hypothesis ($H_0$) – a statistical hypothesis that states that there is no difference between a parameter and a specific value, or that there is no difference between two parameters.

Alternative Hypothesis ($H_1$) – a statistical hypothesis that states the existence of a difference between a parameter and a specific value, or states that there is a difference between two parameters.

$H_0$ will ALWAYS have an equal sign (and possibly a less than or greater than symbol, depending on the alternative hypothesis).

The alternative hypothesis has a range of values that are alternatives to the one in $H_0$.

The null and alternative hypotheses are stated together. The following are typical hypothesis for means, where k is a specified number.

Two-tailed test: $H_0: \mu=k, H_1:\mu\not=k.$

Right-tailed test: $H_0:\mu \leq k, H_1: \mu>k.$

Left-tailed test: $H_0:\mu \geq k, H_1:\mu<k.$

Statistical Test uses the data obtained from a sample to make a decision about whether the null hypothesis should be rejected.

Test Value (test statistic) is the numerical value obtained from a statistical test. 

Type I error: reject $H_0$ when $H_0$ is true.

Type II error: do not reject $H_0$ when $H_0$ is false.

Results of a statistical test:

$$\begin{array}{c:c:c} & H_0\ \text{is True} & H_0\ \text{is False} \\ \hline \text{Reject}\ H_0 & \text{Type I Error} & \text{Correct decision} \\ \hdashline \text{Do not Reject}\ H_0 & \text{Correct decision} & \text{Type II Error} \end{array}$$

Significance level $\alpha$ is the maximum probability of committing a Type I error.

$P(\text{Type I error}|H_0\ \text{is true})=\alpha$

Critical or Rejection Region is the range of values for the test value that indicate a significant difference and that the null hypothesis should be rejected.

Non-critical or Non-rejection Region is the range of values for the test value that indicates that the difference was probably due to chance and that the null hypothesis should not be rejected.

Critical Value $(CV)$ separates the critical region from the non-critical region, i.e., when we should reject $H_0$ from when we should not reject $H_0.$

One-tailed test indicates that the null hypothesis should be rejected when the test value is in the critical region on one side.

Left-tailed test is a test where the critical region is on the left side of the distribution of the test value. 

Right-tailed test is a test where the critical region is on the right side of the distribution of the test value.

Two-tailed test indicates that the null hypothesis should be rejected when the test value is in either of two critical regions on either side of the distribution of the test value.

To obtain the critical value, the researcher must choose the significance level, $\alpha,$ and know the distribution of the test value.

The distribution of the test value indicates the shape of the distribution curve for the test value.

This will have a shape that we know (like the standard normal or $t$ distribution).

Hypothesis Test Procedure (Traditional Method)

Step 1 State the hypotheses and identify the claim.

Step 2 Find the critical value(s) from the appropriate table.

Step 3 Compute the test value.

Step 4 Make the decision to reject or not reject the null hypothesis.

Step 5 Summarize the results.

The $z-$test for Means is a statistical test for the mean of a population. It can be used when $n\geq30,$ or when the population is normally distributed and $\sigma$ is known.

$$Claim\ is\ H_0$$ $$\begin{matrix} Regect\ H_0 & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Do\ not\ reject\ H_0 \\ \end{matrix}$$ $$\begin{matrix} \text{There is enough evidence} &\ \ \ \ \ \ \text{There is enough evidence} \\ \text{to reject the claim}\ \ \ \ \ \ \ \ \ \ \ & \text{to reject the claim} \ \ \ \ \ \end{matrix}$$ $$Claim\ is\ H_1$$ $$\begin{matrix} Regect\ H_0 & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Do\ not\ reject\ H_0 \\ \end{matrix}$$ $$\begin{matrix} \text{There is enough evidence} &\ \ \ \ \ \ \text{There is enough evidence} \\ \text{to support the claim}\ \ \ \ \ \ \ & \text{to support the claim} \ \end{matrix}$$

We will use the $t$ test when $\sigma$ is unknown and the distribution of the variable is approximately normal.

The one-sample $t$ test is a statistical test for the mean of a population and is used when the population is normally or approximately normally distributed and $\sigma$ is unknown. 

P-Value Method for Hypothesis Testing

The P-Value (or probability value) is the probability of getting a sample statistic (such as the mean) or a more extreme sample statistic in the direction of the alternative hypothesis when the null hypothesis is true. The P-value is the actual area under the standard normal distribution curve of the test value or a more extreme value (further in the tail).

A General Rule for Finding P-values using the Z distribution or the t distribution:

Suppose that $z^*$is the test statistic of a $z$ test and $t^*$is the test statistic of a $t$ test

Left-tailed test: $p-value=P(Z\leq z^*)$ or $p-value=P(T\leq t^*)$ (depending on whether we are doing a $z$ test or a $t$ test).

Right-tailed test: $p-value=P(Z\geq z^*)$ or $p-value=P(T\geq t^*)$ (depending on whether we are doing a $z$ test or a $t$ test).

Two-tailed test: p-value $p-value=2P(Z\geq z^*)$ or $p-value=2P(T\geq t^*)$ (depending on whether we are doing a $z$ test or a $t$ test).

The smaller the P-value, the stronger the evidence is against $H_0.$

What level of significance would we like to use?

Steps in hypothesis testing

Step 1 State the hypotheses and identify the claim.

Step 2 Find the critical value(s) from the appropriate table.

Step 3 Compute the test value and determine the P-value.

Step 4 Make the decision to reject or not reject the null hypothesis.

Step 5 Summarize the results.

 z-Test for a Proportion

When both $n \hat{p}$ and $n \hat{q}$ are each greater than or equal to 5, the Central Limit Theorem kicks in and a normal distribution can be used to describe the distribution of the sample proportions. When the central limit theorem applies to data from a binomial distribution, then $\hat{p}$ can be well approximated by a normal distribution with mean $p$ and standard deviation $\sqrt{pq/n}$

When performing our hypothesis test, we will make an assumption about the value of $p$ in our null hypothesis, $H_0:\ p=k,$ where $k$ is some fixed value. We can use this to determine to perform a hypothesis test for $p.$