Unlock Statistical Insights: Confidence Intervals and Hypothesis Testing Explained

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$C I = \overset{x}{ˉ} \pm E$
$\overset{x}{ˉ} - E$	$\overset{x}{ˉ} + E$
$330 - 3.60$	$330 + 3.60$
$326.40$	$333.60$

C I = \overset{x}{ˉ} \pm E

\overset{x}{ˉ} - E

\overset{x}{ˉ} + E

330 - 3.60

330 + 3.60

326.40

333.60

Null Hypothesis	Alternative Hypothesis
The mean is greater than or equal to $0.60 .$ $H_{0} : μ \geq 0.60$	The mean is less than $0.60 .$ (claim) $H_{a} : μ < 0.60$

Null Hypothesis

Alternative Hypothesis

The mean is greater than or equal to

0.60 .

H_{0} : μ \geq 0.60

The mean is less than

0.60 .

(claim)

H_{a} : μ < 0.60

Null Hypothesis	Alternative Hypothesis
The mean is equal to $0.50 .$ (claim) $H_{0} : μ = 0.50$	The mean is not equal to $0.50 .$ $H_{a} : μ \neq = 0.50$

Null Hypothesis

Alternative Hypothesis

The mean is equal to

0.50 .

(claim)

H_{0} : μ = 0.50

The mean is not equal to

0.50 .

H_{a} : μ \neq = 0.50

Typical Claims About the Mean
The mean is equal to a specific value, $μ = k .$	The mean is greater than a specific value, $μ > k .$	The mean is less than a specific value, $μ < k .$

Typical Claims About the Mean

The mean is equal to a specific value,

μ = k .

The mean is greater than a specific value,

μ > k .

The mean is less than a specific value,

μ < k .

Typical Significance Levels $α$
$1 %$	$5 %$	$10 %$

Typical Significance Levels

α

1 %

5 %

10 %

	Critical Values
Significance Level	Left-Tail Test $H_{a} : μ < k$	Two-Tail Test $H_{a} : μ \neq = k$	Right-Tail Test $H_{a} : μ > k$
$α = 1 %$	$- 2.326$	$\pm 2.576$	$2.326$
$α = 5 %$	$- 1.645$	$\pm 1.960$	$1.645$
$α = 10 %$	$- 1.282$	$\pm 1.645$	$1.282$

Critical Values

Significance Level

Left-Tail Test

H_{a} : μ < k

Two-Tail Test

H_{a} : μ \neq = k

Right-Tail Test

H_{a} : μ > k

α = 1 %

- 2.326

\pm 2.576

2.326

α = 5 %

- 1.645

\pm 1.960

1.645

α = 10 %

- 1.282

\pm 1.645

1.282

A company says that each of their packages of ham contains exactly $20$ slices.

Null Hypothesis $H_{0}$	Alternative Hypothesis $H_{a}$
The mean is equal to $20$ slices (claim). $H_{0} : μ = 20$	The mean is different than $20$ slices. $H_{a} : μ \neq = 20$

Null Hypothesis

H_{0}

Alternative Hypothesis

H_{a}

The mean is equal to

20

slices (claim).

H_{0} : μ = 20

The mean is different than

20

slices.

H_{a} : μ \neq = 20

	Critical Values
Significance Level	Left-Tail Test $H_{a} : μ < k$	Right-Tail Test $H_{a} : μ > k$	Two-Tail Test $H_{a} : μ \neq = k$
$α = 1 %$	$- 2.326$	$2.326$	$\pm 2.576$
$α = 5 %$	$- 1.645$	$1.645$	$\pm 1.960$
$α = 10 %$	$- 1.282$	$1.282$	$\pm 1.645$

Critical Values

Significance Level

Left-Tail Test

H_{a} : μ < k

Right-Tail Test

H_{a} : μ > k

Two-Tail Test

H_{a} : μ \neq = k

α = 1 %

- 2.326

2.326

\pm 2.576

α = 5 %

- 1.645

1.645

\pm 1.960

α = 10 %

- 1.282

1.282

\pm 1.645

A company says that each of their packages of ham contains exactly $20$ slices.

Null Hypothesis $H_{0}$	Alternative Hypothesis $H_{a}$
The mean is equal to $150 g .$ (claim) $H_{0} : μ = 150 g$	The mean is different than $150 g .$ $H_{a} : μ \neq = 150 g$

Null Hypothesis

H_{0}

Alternative Hypothesis

H_{a}

The mean is equal to

150 g .

(claim)

H_{0} : μ = 150 g

The mean is different than

150 g .

H_{a} : μ \neq = 150 g

Null Hypothesis $H_{0}$	Alternative Hypothesis $H_{a}$
The mean is less than or equal to $59$ minutes. (claim) $H_{0} : μ \leq 59 minutes$	The mean is greater than $59$ minutes. $H_{a} : μ > 59 minutes$

Null Hypothesis

H_{0}

Alternative Hypothesis

H_{a}

The mean is less than or equal to

59

minutes. (claim)

H_{0} : μ \leq 59 minutes

The mean is greater than

59

minutes.

H_{a} : μ > 59 minutes

Inferential Methods
Confidence Interval	Hypothesis Test
Estimates a population parameter as a range of values	Tests a claim about the mean of a population

Inferential Methods

Confidence Interval

Hypothesis Test

Estimates a population parameter as a range of values

Tests a claim about the mean of a population

Null Hypothesis $H_{0}$	Alternative Hypothesis $H_{a}$
The mean is equal to $24.3 .$ (claim) $H_{0} : μ = 24.3$	The mean is different than $24.3$ years. $H_{a} : μ \neq = 24.3$

Null Hypothesis

H_{0}

Alternative Hypothesis

H_{a}

The mean is equal to

24.3 .

(claim)

H_{0} : μ = 24.3

The mean is different than

24.3

years.

H_{a} : μ \neq = 24.3

A laptop manufacturing company says it takes no more than $48$ minutes for the battery to fully charge. Using a sample of $38$ laptops, Ali calculated a mean time of $53$ minutes with a standard deviation of $10.45$ minutes.

A laptop with its battery fully charged.

If a test with $10 %$ significance is performed, analyze the following situations and select the ones that fit this study.

Select the test of significance that fits this study.

Consider the following graphs.

Select the graph that represents the critical region for this hypothesis test.

What is the value of the

z -

statistic? Round to three decimal places.

Consider the following statements.

A . B . The mean time it takes the laptop battery to charge fully is no more than 48 minutes . The mean time it takes the laptop battery to charge fully is more than 48 minutes .

Which statement is more likely true?

We will identify if a left-tailed, right-tailed, or two-tailed test fits this situation. To do so, let's begin by identifying the claim to set the null and alternative hypotheses.

It takes no more than 48 minutes for the laptop battery to charge fully.

We can represent the company's claim with a non-strict inequality. μ≤48 Because of the ≤ sign, we can relate the claim to the null hypothesis. Conversely, the complement will be the alternative hypothesis H_a.

Null Hypothesis H_0	Alternative Hypothesis H_a
The mean is less than or equal to 48 minutes. (claim) H_0: μ≤48 minutes	The mean is greater than 48 minutes. H_a:μ> 48 minutes

Now we can identify the inequality symbol of the alternative hypothesis to see which test of significance applies to the given situation. Since the inequality symbol is >, a right-tailed test fits this situation.

We are performing a right-tailed test at 10 % significance so that the critical region will be in the upper 10 % of the standard normal distribution. We will calculate the z-value of the upper 10 % to find the critical value and set the critical region. We will use a graphing calculator. Push 2nd, then VARS, and select the third option, invNorm(.

Because we want the upper 10 % of the distribution, the value to be entered into the calculator is given by 1-0.1=0.9. Next, we will enter this value and push ENTER to get the result

The critical value is about 1.282. We can now set the critical region in the upper tail of the distribution by using this value.

Note that this graph corresponds to option A.

We will now calculate the z-statistic using the following formula. z=x-μ/ssqrt(n) In this formula, x is the sample mean, μ is the population mean, s is the standard deviation of the sample, and n is the sample size. In this case, we are given that x= 53, μ= 48, s= 10.45, and n= 38.

Now that we have the z-statistic, we can plot it on the graph of the critical region to see its position in the distribution. If it falls in the critical region, we can reject the null hypothesis. Let's do it!

We can see that the z-statistic is in the critical region, so we can reject the null hypothesis. Recall that the initial claim is related to the null hypothesis.

It takes no more than 48 minutes for the laptop battery to charge fully.

Therefore, we can reject the claim and say that it is more likely that the mean time it takes the laptop battery to charge fully is more than 48 minutes. This is statement B.

A famous gaming chair manufacturer claims that the chairs they produce last for at least ten years. Suppose that from a sample of $33$ chairs, Davontay found that the mean time that the sample chairs lasted was $9.5$ years with a standard deviation of $1.5$ years.

If a $1 %$ hypothesis test is conducted using this sample data, determine the information corresponding to the following situations.

Which test of significance will be used to test the manufacturer's claim?

What is the critical value for the

1 %

significance level? Round to three decimal places.

What is the value of the

z -

statistic? Round to three decimal places.

Consider the following statements.

A . B . The gaming chairs made by this company last for at least 10 years . The gaming chairs made by this company last for less than 10 years .

Which statement is more likely true?

To identify the test of significance that applies to this situation, we need first to analyze the company's claim to set the null and alternative hypotheses. Then according to the inequality sign of the alternative hypothesis, we can determine the test of significance. Let's take a look at the company's claim.

Our gaming chairs last for at least 10 years.

We can represent the claim of the manufacturer company as the following non-strict inequality. μ≥ 10 We can see that the inequality symbol in this expression is ≥, which means we can relate the claim to the null hypothesis. The alternative hypothesis will be the complement of this statement.

Null Hypothesis H_0	Alternative Hypothesis H_a
The mean is greater than or equal to 10 years. (claim) H_0: μ≥10 years	The mean is less than 10 years. H_a:μ< 10 years

Note that the inequality sign of the alternative hypothesis is <. Therefore, a left-tailed test is needed when making a hypothesis test to evaluate the company's claim.

Given that we will make a left-tailed test and the significance level is 1 %, we need to find the critical value that separates the 1 % of the area in the left tail of the standard normal distribution. To do so, we will use a graphing calculator. Push 2nd, then VARS, and select the third option, invNorm(.

Next, we will enter 0.01 and push ENTER to get the critical value corresponding to the 1 %.

The critical value is about -2.326. By using this value, we can draw the critical region for this hypothesis test.

Consider the formula for the z-statistic. z=x-μ/ssqrt(n) In this formula, x is the sample mean, μ is the population mean, s is the standard deviation of the sample, and n is the sample size. In this case, we are told that x= 9.5, μ= 10, s= 1.5, and n= 33.

Now let's plot the z-statistic to see if it falls in the critical region. If so, we will reject the null hypothesis. If it does not, we cannot reject the null hypothesis.

Because the z-statistic falls outside the critical region, we cannot reject the null hypothesis. Let's recall the claim of the gaming chair manufacturing, which is related to the null hypothesis.

Our gaming chairs last for at least 10 years.

Since we cannot reject the claim, it means that the company statement that the gaming chairs last for at least ten years is most likely true.

A company that packages cereal wants to test if their machines are functioning well. The machines must package the correct weight in each box they sell. If the weight of the boxes are off, the company is either shorting customers or losing money.

The company claims that the mean weight of the boxes is

510

grams. A sample of

40

boxes of cereal packaged last week by this machine has a mean weight of

516

grams with a standard deviation of

20 .

Consider the following statements.

A . B . The machine is functioning well . The machine is malfunctioning .

Suppose a hypothesis test with a

10 %

significance level is performed. Which statement is more likely true about the machine?

We will make a hypothesis test to identify which statement is more likely true about the machine that packages the cereal boxes. Let's recall the five steps to perform a hypothesis test.

Identify the claim and state the null and alternative hypothesis.
Determine the critical value(s) and critical region(s).
Calculate the z-statistic.
Reject or fail to reject the null hypothesis.
Make a conclusion about the claim.

Let's follow these steps one at a time.

Identify the Claim and State the Null and Alternative Hypotheses

We are told that the company claims that the mean weight of cereal boxes equals 510 grams. We will represent this claim by the following equality. μ=510 Because this is a statement of equality, it will represent the null hypothesis. Conversely, the alternative hypothesis is that the mean weight of cereal boxes is not 510 grams.

Null Hypothesis H_0	Alternative Hypothesis H_a
The mean equals 510 grams. (claim) H_0: μ=510 grams	The mean is not equal to 510 grams. H_a:μ≠ 510 grams

Determine the Critical Value(s) and Critical Region(s)

Since the sign of the alternative hypothesis is ≠, we will conduct a two-tailed test of significance. This means that there are two critical regions whose cutoffs will be given by the z-value of the significance level α. The following are the critical values for the most common α values.

	Critical Values
Significance Level	Left-Tail Test H_a:μ	Right-Tail Test H_a:μ>k	Two-Tail Test H_a:μ≠ k
α=1 %	-2.326	2.326	±2.576
α=5 %	-1.645	1.645	±1.960
α=10 %	-1.282	1.282	±1.645

From the table, we can see that the critical values for a 10 % significance level are ±1.645. Let's use these values to label the critical regions and critical values in a standard normal distribution.

Calculate the z-statistic

Recall that the z-statistic is the z-value of the sample mean and can be calculated by the following formula. z=x-μ/ssqrt(n) In this formula, x is the sample mean, μ is the population mean, s is the standard deviation of the sample, and n is the sample size. For the given situation, we have that x= 516, μ= 510, s= 20, and n= 40.

Reject or Fail to Reject the Null Hypothesis

We will now verify if the z-statistic falls in the critical region. If so, we will reject the null hypothesis. Let's plot the z-statistic into the graph of the critical regions.

Because the z-statistic is in the critical region, we must reject the null hypothesis H_0.

Make a Conclusion About the Claim

Knowing that there is evidence to reject the null hypothesis, let's now make a conclusion about the initial claim.

The mean weight of cereal boxes equals 510 grams.

Because the initial claim is related to the null hypothesis, we can reject the claim that the boxes weigh, on average, 510 grams. This implies that the machine packaging the boxes is malfunctioning, so statement B is true.

	11 Theory slides
	8 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Confidence Intervals and Hypothesis Testing

Catch-Up and Review

Confidence Level and the Standard Normal Distribution

Confidence Interval for the Population Mean

Hint

Solution

Finding the z-value

Evaluating the Formula

Hint

Solution

Calculating the Maximum Error of Estimate

Determining the Confidence Interval

Example

Hint

Solution

Extra

Hint

Solution

Hint

Solution

Hint

Solution

Identify the Claim and State the Null and Alternative Hypotheses

Determine the Critical Value(s) and Critical Region(s)

Calculate the z-statistic

Reject or Fail to Reject the Null Hypothesis

Make a Conclusion About the Claim

Confidence Intervals and Hypothesis Testing

Recommended exercises

Finding the $z -$ value