Hypothesis testing with two samples | Practice Problems, Examples & Solutions

Two Population Means with Unknown Standard Deviations

Statistics Informed Decisions Using Data

We often hear sports announcers say, "I wonder which player will show up to play today." This is the announcer's way of saying that the player is inconsistent, that his or her performance varies dramatically from game to game. Suppose that the standard deviation of the number of points scored by shooting guards in the NBA is $8.3 . \mathrm{A}$ random sample of 25 games played by Derrick Rose results in a sample standard deviation of 6.7 points. Assume that a normal probability plot indicates that the points scored are approximately normally distributed. Is Derrick Rose more consistent than other shooting guards in the NBA at the $\alpha=0.10$ level of significance?

Hypothesis Tests Regarding a Parameter

Hypothesis Tests for a Population Standard Deviation

02:05

Statistics Informed Decisions Using Data

A machine fills bottles with 64 fluid ounces of liquid. The quality-control manager determines that the fill levels are normally distributed with a mean of 64 ounces and a standard deviation of 0.42 ounce. He has an engineer recalibrate the machine in an attempt to lower the standard deviation. After the recalibration, the quality-control manager randomly selects 19 bottles from the line and determines that the standard deviation is 0.38 ounce. Is there less variability in the filling machine? Use the $\alpha=0.01$ level of significance.

Hypothesis Tests Regarding a Parameter

Hypothesis Tests for a Population Standard Deviation

02:07

Statistics Informed Decisions Using Data

. To test $H_{0}: \sigma=0.35$ versus $H_{1}: \sigma<0.35,$ a random sample of size $n=41$ is obtained from a population that is known to be normally distributed.
(a) If the sample standard deviation is determined to be $s=0.23$ compute the test statistic.
(b) If the researcher decides to test this hypothesis at the $\alpha=0.01$ level of significance, determine the critical value.
(c) Draw a chi-square distribution and depict the critical region.
(d) Will the researcher reject the null hypothesis? Why?

Hypothesis Tests Regarding a Parameter

Hypothesis Tests for a Population Standard Deviation

Two Population Means with Known Standard Deviations

11 Practice Problems

14:25

Statistics for Business and Economics

Gasoline prices reached record high levels in 16 states during 2003 (The Wall Street Journal, March 7,2003 ). Two of the affected states were California and Florida. The American Automobile Association reported a sample mean price of $\$ 2.04$ per gallon in California and a sample mean price of $\$ 1.72$ per gallon in Florida. Use a sample size of 40 for the California data and a sample size of 35 for the Florida data. Assume that prior studies indicate a population standard deviation of .10 in California and .08 in Florida are reasonable.
a. What is a point estimate of the difference between the population mean prices per gallon in California and Florida?
b. At $95 \%$ confidence, what is the margin of error?
c. What is the $95 \%$ confidence interval estimate of the difference between the population mean prices per gallon in the two states?

Statistical Inference About Means and Proportions with Two Populations

02:03

Statistics Informed Decisions Using Data

Why do we use a pooled estimate of the population proportion when testing a hypothesis about two proportions? Why do we not use a pooled estimate of the population proportion when constructing a confidence interval for the difference of two proportions?

Inferences on Two Samples

Inference about Two Population Proportions

02:13

Statistics Informed Decisions Using Data

Voice-Recognition Systems Have you ever been frustrated by computer telephone systems that do not understand your voice commands? Quite a bit of effort goes into designing these systems to minimize voice-recognition errors. Researchers at the Oregon Graduate Institute of Science and Technology developed a new method of voice recognition (called a remapped network) that was thought to be an improvement over an existing neural network. The data shown are based on results of their research. Does the evidence suggest that the remapped network has a different proportion of errors than the neural network? Use the $\alpha=0.05$ level of significance.
TABLE CANT COPY

Inferences on Two Samples

Inference about Two Population Proportions

Comparing Two Independent Population Proportions

11 Practice Problems

05:24

Modern Physics

Cavity modes in lasers. Under high resolution and at threshold current, the emission spectrum from a GaAs laser is seen to consist of many sharp lines, as shown in Figure P12.19. Although one line generally dominates at higher currents, let us consider the origin and spacing of the multiple lines, or modes, shown in this figure. Recall that standing waves or resonant modes corresponding to the sharp emission lines are formed when an integral number $m$ of half-wavelengths fits between the cleaved GaAs surfaces. If $L$ is the distance between cleaved faces, $n$ is the index of refraction of the semiconductor, and $\lambda$ is the wavelength in air, we have Figure P12.19 A high-resolution emission spectrum of the GaAs laser, operated just below the laser threshold. $$m=\frac{L}{\lambda / 2 n}$$ (a) Show that the wavelength separation between adjacent modes (the change in wavelength for the case $\Delta m=+1)$ is $$|\Delta \lambda|=\frac{\lambda^{2}}{2 L\left(n-\lambda \frac{d n}{d \lambda}\right)}$$ (Hint: since $m$ is large and we want the small change in wavelength that corresponds to $\Delta m=1,$ we can consider $m$ to be a continuous function of $\lambda$ and differentiate $m=2 L n / \lambda) .$ (b) Using the following typical values for GaAs, calculate the wavelength separation between adjacent modes. Compare your results to Figure P12.19. $$\begin{aligned}
\lambda &=837 \mathrm{nm} \\
2 L &=0.60 \mathrm{nm} \\
n &=3.58 \\
n / d \lambda &\left.=3.8 \times 10^{-4}(\mathrm{nm})^{-1}\right\} \mathrm{at} \lambda=837 \mathrm{nm}
\end{aligned}$$ (c) Estimate the mode separation for the He-Ne gas laser using $\lambda=632.8 \mathrm{nm}, 2 L=0.6 \mathrm{m}, n \approx 1,$ and
$d n / d \lambda=0 .$ On the basis of this calculation and the result of part (b), what is the controlling factor in mode separation in both solids and gases?

The Solid State

Testing $\mu_{1}-\mu_{2}$ and $p_{1}-p_{2}$ (Independent Samples)

02:53

Understandable Statistics, Concepts and Methods

Please provide the following information for Problems.
(a) What is the level of significance? State the null and alternate hypotheses.
(b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding $z$ or $t$ value as appropriate.
(c) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value.
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level $\alpha ?$
(e) Interpret your conclusion in the context of the application. Note: For degrees of freedom $d . f .$ not in the Student's $t$ table, use the closest $d . f .$ that is smaller. In some situations, this choice of $d . f .$ may increase the $P$ -value a small amount and therefore produce a slightly more "conservative" answer.
Sociology: High School Dropouts This problem is based on information taken from Life in America's Fifty States by G. S. Thomas. A random sample of $n_{1}=153$ people ages 16 to 19 was taken from the island of Oahu, Hawaii, and 12 were found to be high school dropouts. Another random sample of $n_{2}=128$ people ages 16 to 19 was taken from Sweetwater County, Wyoming, and 7 were found to be high school dropouts. Do these data indicate that the population proportion of high school dropouts on Oahu is different (either way) from that of Sweetwater County? Use a
$1 \%$ level of significance.

Hypothesis Testing

Testing $\mu_{1}-\mu_{2}$ and $p_{1}-p_{2}$ (Independent Samples)

04:26

Understandable Statistics, Concepts and Methods

Please provide the following information for Problems.
(a) What is the level of significance? State the null and alternate hypotheses.
(b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding $z$ or $t$ value as appropriate.
(c) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value.
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level $\alpha ?$
(e) Interpret your conclusion in the context of the application. Note: For degrees of freedom $d . f .$ not in the Student's $t$ table, use the closest $d . f .$ that is smaller. In some situations, this choice of $d . f .$ may increase the $P$ -value a small amount and therefore produce a slightly more "conservative" answer.
Education: Tutoring In the journal Mental Retardation, an article reported the results of a peer tutoring program to help mildly mentally retarded children learn to read. In the experiment, the mildly retarded children were randomly divided into two groups: the experimental group received peer tutoring along with regular instruction, and the control group received regular instruction with no peer tutoring. There were $n_{1}=n_{2}=30$ children in each group. The Gates-MacGintie Reading Test was given to both groups before instruction began. For the experimental group, the mean score on the vocabulary portion of the test was $\bar{x}_{1}=344.5,$ with sample standard deviation $s_{1}=49.1$ For the control group, the mean score on the same test was $\bar{x}_{2}=354.2,$ with sample standard deviation $s_{2}=50.9 .$ Use a $5 \%$ level of significance to test the hypothesis that there was no difference in the vocabulary scores of the I two groups before the instruction began.

Hypothesis Testing

Testing $\mu_{1}-\mu_{2}$ and $p_{1}-p_{2}$ (Independent Samples)

Matched or Paired Samples

11 Practice Problems

01:01

Statistics for Business Economics

The manager of the Danvers-Hilton Resort Hotel stated that the mean guest bill for a weekend is $\$ 600$ or less. A member of the hotel's accounting staff noticed that the total charges for guest bills have been increasing in recent months. The accountant will use a sample of weekend guest bills to test the manager's claim.
a. Which form of the hypotheses should be used to test the manager's claim? Explain.
\[
\begin{array}{lll}
H_{0^{*}} \mu \geq 600 & H_{0^{*}} \mu \leq 600 & H_{0^{*}} \mu=600 \\
H_{\mathrm{a}^{*}} \mu<600 & H_{\mathrm{a}^{*}} \mu>600 & H_{\mathrm{a}^{*}} \mu \neq 600
\end{array}
\]
b. What conclusion is appropriate when $H_{0}$ cannot be rejected?
c. What conclusion is appropriate when $H_{0}$ can be rejected?

Hypothesis Tests

01:37

Statistics Informed Decisions Using Data

Quality Control Suppose the mean wait-time for a telephone reservation agent at a large airline is 43 seconds. A manager with the airline is concerned that business may be lost due to customers having to wait too long for an agent. To address this concern, the manager develops new airline reservation policies that are intended to reduce the amount of time an agent needs to spend with each customer. A random sample of 250 customers results in a sample mean wait-time of 42.3 seconds with a standard deviation of 4.2 seconds. Using an $\alpha=0.05$ level of significance, do you believe the new policies were effective? Do you think the results have any practical significance?

Hypothesis Tests Regarding a Parameter

Putting It Together: Which Method Do I Use?

02:12

Statistics Informed Decisions Using Data

Toner Cartridge The manufacturer of a toner cartridge claims the mean number of printouts is 10,000 for each cartridge. A consumer advocate is concerned that the actual mean number of printouts is lower. He selects a random sample of 14 such cartridges and obtains the following number of printouts: $$\begin{array}{lllllll} 9,600 & 10,300 & 9,000 & 10,750 & 9,490 & 9,080 & 9,655 \\ \hline 9,520 & 10,070 & 9,999 & 10,470 & 8,920 & 9,964 & 10,330 \end{array}$$
(a) Because the sample size is small, he must verify that the number of printouts is normally distributed and the sample does not contain any outliers. The normal probability plot and boxplot are shown. Are the conditions for testing the hypothesis satisfied? (FIGURE CAN'T COPY)
(b) Are the consumer advocate's concerns founded? Use the $\alpha=0.05$ level of significance.

Hypothesis Tests Regarding a Parameter

Putting It Together: Which Method Do I Use?

Hypothesis Testing for Two Means and Two Proportions

18 Practice Problems

00:29

21st Century Astronomy

An empirical science is one that is based on
a. hypothesis.
b. calculus.
c. computer models.
d. observed data.

Motion of Astronomical Bodies

03:00

Elementary Statistics

No Variation in a Sample An experiment was conducted to test the effects of alcohol. Researchers measured the breath alcohol levels for a treatment group of people who drank ethanol and another group given a placebo. The results are given below (based on data from "Effects of Alcohol Intoxication on Risk Taking, Strategy, and Error Rate in Visuomotor Performance," by Streufert et al., Joumal of Applied Psychology, Vol. 77, No. 4). Use a 0.05 significance level to test the claim that the two sample groups come from populations with the same mean.
$$\begin{aligned}
&\text { Treatment Group: } \quad n_{1}=22, \bar{x}_{1}=0.049, s_{1}=0.015\\
&\text { Placebo Group: } \quad n_{2}=22, \bar{x}_{2}=0.000, s_{2}=0.000
\end{aligned}$$

Inferences from Two Samples

Two Means: Independent Samples

02:33

Elementary Statistics

Pooling Repeat Exercise 12 "IQ and Lead" by assuming that the two population standard deviations are equal, so $\sigma_{1}=\sigma_{2}$. Use the appropriate method from Part 2 of this section. Does pooling the standard deviations yield results showing greater significance?

Inferences from Two Samples

Two Means: Independent Samples

Testing of Hypothesis

53 Practice Problems

01:20

Statistics Informed Decisions Using Data

Redo Problem $3(b)$ with $\alpha=0.01 .$ What effect does lowering the level of significance have on the power of the test? Why does this make sense?

Hypothesis Tests Regarding a Parameter

The Probability of a Type II Error and the Power of the Test

04:12

Statistics Informed Decisions Using Data

In August $2002,47 \%$ of parents who had children in grades $K-12$ were satisfied with the quality of education the students receive. In September 2010 , the Gallup organization conducted a poll of 1013 parents who had children in grades $\mathrm{K}-12$ and asked if they were satisfied with the quality of education the students receive. Of the 1013 surveyed, 437 indicated they were satisfied. Does this suggest the proportion of parents satisfied with the quality of education has decreased?
(a) What does it mean to make a Type II error for this test?
(b) If the researcher decides to test this hypothesis at the $\alpha=0.10$ level of significance, determine the probability of making a Type II error if the true population proportion is $0.42 .$ What is the power of the test?
(c) Redo part (b) if the true proportion is 0.46.

Hypothesis Tests Regarding a Parameter

The Probability of a Type II Error and the Power of the Test

03:53

Statistics Informed Decisions Using Data

To test $H_{0}: p=0.75$ versus $H_{1}: p<0.75,$ a simple random sample of $n=400$ individuals is obtained and $x=280$ successes are observed.
(a) What does it mean to make a Type II error for this test?
(b) If the researcher decides to test this hypothesis at the $\alpha=0.01$ level of significance, compute the probability of making a Type II error if the true population proportion is $0.71 .$ What is the power of the test?
(c) Redo part (b) if the true population proportion is 0.68.

Hypothesis Tests Regarding a Parameter

The Probability of a Type II Error and the Power of the Test

Types of Errors, Level of Significance and p-value

26 Practice Problems

02:36

Elementary Statistics

According to Benford's law, a variety of different data sets include numbers with leading (first) digits that follow the distribution shown in the table below.Test for goodness-of-fit with the distribution described by Benford's law.
$$\begin{array}{l|c|c|c|c|c|c|c|c|c}
\hline \text { Leading Digit } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline \begin{array}{l}
\text { Benford's Law: Distribution } \\
\text { of Leading Digits }
\end{array} & 30.1 \% & 17.6 \% & 12.5 \% & 9.7 \% & 7.9 \% & 6.7 \% & 5.8 \% & 5.1 \% & 4.6 \% \\
\hline
\end{array}$$
Frequencies of leading digits from IRS tax files are 152,89,63,48,39,40
$28,25,$ and 27 (corresponding to the leading digits of $1,2,3,4,5,6,7,8,$ and $9,$ respectively, based on data from Mark Nigrini, who provides software for Benford data analysis). Using a 0.05 significance level, test for goodness-of-fit with Benford's law. Does it appear that the tax entries are legitimate?

Goodness-of-Fit and Contingency Tables

Goodness-of-Fit

02:50

Elementary Statistics

According to Benford's law, a variety of different data sets include numbers with leading (first) digits that follow the distribution shown in the table below.Test for goodness-of-fit with the distribution described by Benford's law.
$$\begin{array}{l|c|c|c|c|c|c|c|c|c}
\hline \text { Leading Digit } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline \begin{array}{l}
\text { Benford's Law: Distribution } \\
\text { of Leading Digits }
\end{array} & 30.1 \% & 17.6 \% & 12.5 \% & 9.7 \% & 7.9 \% & 6.7 \% & 5.8 \% & 5.1 \% & 4.6 \% \\
\hline
\end{array}$$
When working for the Brooklyn district attorney, investigator Robert Burton analyzed the leading digits of the amounts from 784 checks issued by seven suspect companies. The frequencies were found to be $0,15,0,76,479,183,8,23,$ and $0,$ and those digits correspond to the leading digits of $1,2,3,4,5,6,7,8,$ and $9,$ respectively. If the observed frequencies are substantially different from the frequencies expected with Benford's law, the check amounts appear to result from fraud. Use a 0.01 significance level to test for goodnessof-fit with Benford's law. Does it appear that the checks are the result of fraud?

Goodness-of-Fit and Contingency Tables

Goodness-of-Fit

15:54

Elementary Statistics

Conduct the hypothesis test and provide the test statistic and the P-value and/or critical value, and state the conclusion.
The table below lists the numbers of games played in 105 Major League Baseball (MLB) World Series. This table also includes the expected proportions for the numbers of games in a World Series, assuming that in each series, both teams have about the same chance of winning. Use a 0.05 significance level to test the claim that the actual numbers of games fit the distribution indicated by the expected proportions.
$$\begin{array}{l|c|c|c|c}
\hline \text { Games Played } & 4 & 5 & 6 & 7 \\
\hline \text { World Series Contests } & 21 & 23 & 23 & 38 \\
\hline \text { Expected Proportion } & 2 / 16 & 4 / 16 & 5 / 16 & 5 / 16 \\
\hline
\end{array}$$

Goodness-of-Fit and Contingency Tables

Goodness-of-Fit

Chi-Square Test

0 Practice Problems

Independent and Dependent Samples

17 Practice Problems

02:51

Understandable Statistics, Concepts and Methods

Please provide the following information for Problems.
(a) What is the level of significance? State the null and alternate hypotheses.
(b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding $z$ or $t$ value as appropriate.
(c) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value.
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level $\alpha ?$
(e) Interpret your conclusion in the context of the application. Note: For degrees of freedom $d . f .$ not in the Student's $t$ table, use the closest $d . f .$ that is smaller. In some situations, this choice of $d . f .$ may increase the $P$ -value a small amount and therefore produce a slightly more "conservative" answer.
Management: Intimidators and Stressors This problem is based on information regarding productivity in leading Silicon Valley companies (see reference in Problem 25 ). In large corporations, an "intimidator" is an employee who tries to stop communication, sometimes sabotages others, and, above all, likes to listen to him- or herself talk. Let $x_{1}$ be a random variable representing productive hours per week lost by peer employees of an intimidator.
A "stressor" is an employee with a hot temper that leads to unproductive tantrums in corporate society. Let $x_{2}$ be a random variable representing productive hours per week lost by peer employees of a stressor.
i. Use a calculator with mean and standard deviation keys to verify that $\bar{x}_{1}=4.00, s_{1} \approx 2.38, \bar{x}_{2}=5.5,$ and $s_{2} \approx 2.78$
ii. Assuming the variables $x_{1}$ and $x_{2}$ are independent, do the data indicate that the population mean time lost due to stressors is greater than the population mean time lost due to intimidators? Use a $5 \%$ level of significance. (Assume the population distributions of time lost due to intimidators and time lost due to stressors are each mound-shaped and symmetric.

Hypothesis Testing

Testing $\mu_{1}-\mu_{2}$ and $p_{1}-p_{2}$ (Independent Samples)

08:43

Understandable Statistics, Concepts and Methods

A random sample of 49 measurements from one population had a sample mean of $10,$ with sample standard deviation $3 .$ An independent random sample of 64 measurements from a second population had a sample mean of $12,$ with sample standard deviation $4 .$ Test the claim that the population means are different. Use level of significance 0.01.
(a) Check Requirements What distribution does the sample test statistic follow? Explain.
(b) State the hypotheses.
(c) Compute $\bar{x}_{1}-\bar{x}_{2}$ and the corresponding sample distribution value.
(d) Estimate the $P$ -value of the sample test statistic.
(e) Conclude the test.
(f) Interpret the results.

Hypothesis Testing

Testing $\mu_{1}-\mu_{2}$ and $p_{1}-p_{2}$ (Independent Samples)

04:32

Understandable Statistics, Concepts and Methods

Consider a hypothesis test of difference of means for two independent populations $x_{1}$ and $x_{2}$
(a) What does the null hypothesis say about the relationship between the two population means?
(b) If the sample test statistic has a $z$ distribution, give the formula for $z$
(c) If the sample test statistic has a $t$ distribution, give the formula for $t$.

Hypothesis Testing

Testing $\mu_{1}-\mu_{2}$ and $p_{1}-p_{2}$ (Independent Samples)

z-test

14 Practice Problems

01:12

Elementary Statistics: Picturing the World

(a) identify the claim and state $H_{0}$, and $H_{a},(b)$ find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
A guidance counselor claims that high school students in a college preparation program have higher ACT scores than those in a general program. The mean ACT score for 49 high school students who are in a college preparation program is $22.2 .$ Assume the population standard deviation is $4.8 .$ The mean ACT score for 44 high school students who are in a general program
is $20.0 .$ Assume the population standard deviation is $5.4 .$ At $\alpha=0.10,$ can you support the guidance counselor's claim?

Hypothesis Testing with Two Samples

Testing the Difference Between Means (Independent Samples, $\sigma_{1}$ and $\sigma_{2}$ Known)

01:26

Elementary Statistics: Picturing the World

(a) identify the claim and state $H_{0}$, and $H_{a},(b)$ find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
An energy company wants to choose between two regions in a state to install energy-producing wind turbines. A researcher claims that the wind speed in Region $A$ is less than the wind speed in Region B. To test the regions, the average wind speed is calculated for 60 days in each region. The mean wind speed in Region $A$ is 14.0 miles per hour. Assume the population standard deviation is 2.9 miles per hour. The mean wind speed in Region $\mathrm{B}$ is 15.1 miles per hour. Assume the population standard deviation is 3.3 miles per hour. At $\alpha=0.05,$ can the company support the researcher's claim?

Hypothesis Testing with Two Samples

Testing the Difference Between Means (Independent Samples, $\sigma_{1}$ and $\sigma_{2}$ Known)

01:19

Elementary Statistics: Picturing the World

(a) identify the claim and state $H_{0}$, and $H_{a},(b)$ find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed. If convenient, use technology.
To compare the braking distances for two types of tires, a safety engineer conducts 35 braking tests for each type. The mean braking distance for Type $A$ is 42 feet. Assume the population standard deviation is 4.7 feet. The mean braking distance for Type $\mathrm{B}$ is 45 feet. Assume the population standard deviation is 4.3 feet. At $\alpha=0.10,$ can the engineer support the claim that the mean braking distances are different for the two types of tires?

Hypothesis Testing with Two Samples

Testing the Difference Between Means (Independent Samples, $\sigma_{1}$ and $\sigma_{2}$ Known)

Null and Alternative Hypothesis

19 Practice Problems

01:23

Statistics Informed Decisions Using Data

Test the hypothesis using (a) the classical approach and (b) the P-value approach. Be sure to verify the requirements of the test.
$$\begin{aligned}
&H_{0}: p=0.9 \text { versus } H_{1}: p \neq 0.9\\
&n=500 ; x=440 ; \alpha=0.05
\end{aligned}$$

Hypothesis Tests Regarding a Parameter

Hypothesis Tests for a Population Proportion

01:09

Statistics Informed Decisions Using Data

When observed results are unlikely under the assumption that the null hypothesis is true, we say the result is _____ and we reject the null hypothesis.

Hypothesis Tests Regarding a Parameter

Hypothesis Tests for a Population Proportion

03:58

Elementary Statistics

When using the data from Exercise 1 to test for goodness-of-fit with the distribution described by Benford's law, identify the null and alternative hypotheses.

Goodness-of-Fit and Contingency Tables

Goodness-of-Fit