MATH 225N WEEK 8 Final Exam 2025-2026 (MAY-AUGUST QTR) (Version 1&2), Exams of Mathematics

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MATH 225N WEEK 8 Final Exam 202 5 /202 6 (MAY-AUGUST QTR)

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MATH 225N Week 8 Final Exam

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Question 1 1/1 points A fitness center claims that the mean amount of time that a person spends at the gym per visit is 33 minutes. Identify the null hypothesis, H 0, and the alternative hypothesis, Ha , in terms of the parameter μ. That is correct! H 0: μ ≠33; Ha : μ = H 0: μ =33; Ha : μ ≠ H 0: μ ≥33; Ha : μ < H 0: μ ≤33; Ha : μ > Answer Explanation Correct answer: H 0: μ =33; Ha : μ ≠ Let the parameter μ be used to represent the mean. The null hypothesis is always stated with some form of equality: equal (=), greater than or equal to (≥), or less than or equal to (≤). Therefore, in this case, the null hypothesis H 0 is μ =33. The alternative hypothesis is contradictory to the null hypothesis, so Ha is μ ≠33. Question 2 1/1 points The answer choices below represent different hypothesis tests. Which of the choices are right- tailed tests? Select all correct answers.

That is correct!

• H 0: X ≥17.1, Ha : X <17.
• H 0: X =14.4, Ha : X ≠14.
• H 0: X ≤3.8, Ha : X >3.
• H 0: X ≤7.4, Ha : X >7.
• H 0: X =3.3, Ha : X ≠3.

Answer Explanation Correct answer: H 0: X ≤3.8, Ha : X >3. H 0: X ≤7.4, Ha : X >7.

Remember the forms of the hypothesis tests.

• Right-tailed: H 0: X ≤ X 0, Ha : X > X 0.
• Left-tailed: H 0: X ≥ X 0, Ha : X < X 0.
• Two-tailed: H 0: X = X 0, Ha : X ≠ X 0. So in this case, the right-tailed tests are:
• H 0: X ≤7.4, Ha : X >7.
• H 0: X ≤3.8, Ha : X >3. Question 3 1/1 points Find the Type II error given that the null hypothesis, H 0, is: a building inspector claims that no more than 15% of structures in the county were built without permits. That is correct! The building inspector thinks that no more than 15% of the structures in the county were built without permits when, in fact, no more than 15% of the structures really were built without permits. The building inspector thinks that more than 15% of the structures in the county were built without permits when, in fact, more than 15% of the structures really were built without permits. The building inspector thinks that more than 15% of the structures in the county were built without permits when, in fact, at most 15% of the structures were built without permits. The building inspector thinks that no more than 15% of the structures in the county were built without permits when, in fact, more than 15% of the structures were built without permits. Answer Explanation Correct answer: The building inspector thinks that no more than 15% of the structures in the county were built without permits when, in fact, more than 15% of the structures were built without permits. A Type II error is the decision not to reject the null hypothesis when, in fact, it is false. In this case, the Type II error is when the building inspector thinks that no more than 15% of the

structures were built without permits when, in fact, more than 15% of the structures were built without permits.

Question 4 1/1 points Suppose a chef claims that her meatball weight is less than 4 ounces, on average. Several of her customers do not believe her, so the chef decides to do a hypothesis test, at a 10% significance level, to persuade them. She cooks 14 meatballs. The mean weight of the sample meatballs is 3. ounces. The chef knows from experience that the standard deviation for her meatball weight is ounces.

• H 0: μ ≥4; Ha : μ <
• α =0.1 (significance level) What is the test statistic ( z - score) of this one-mean hypothesis test, rounded to two decimal places? That is correct! Test statistic = minus 2 point 2 4$$ Test statistic = minus 2 point 2 4 - correct Answer Explanation Correct answers:
• Test statistic = minus 2 point 2 4 $\text{Test statistic = }-2.24$

The hypotheses were chosen, and the significance level was decided on, so the next step in hypothesis testing is to compute the test statistic. In this scenario, the sample mean weight, x ¯=3.7. The sample the chef uses is 14 meatballs, so n =14. She knows the standard deviation of the meatballs, σ =0.5. Lastly, the chef is comparing the population mean weight to 4 ounces. So, this value (found in the null and alternative hypotheses) is μ 0. Now we will substitute the values into the formula to compute the test statistic: z 0= x ¯− μ 0 σn √=3.7−40.514√≈−0.30.134≈−2. So, the test statistic for this hypothesis test is z 0=−2.24.

Question 5 1/1 points What is the p - value of a right-tailed one-mean hypothesis test, with a test statistic of z 0=1.74? (Do not round your answer; compute your answer using a value from the table below.) z1.51.61.71.81.90.00 0.9330.9450.9550.9640.971 0.01 0.9340.9460.9560.9650.972 0.02 0.9360. 0.9570.9660.973 0.03 0.9370.9480.9580.9660.973 0.04 0.9380.9490.9590.9670.974 0.05 0.9390. 0.9600.9680.974 0.06 0.9410.9520.9610.9690.975 0.07 0.9420.9530.9620.9690.976 0.08 0.9430. 0.9620.9700.976 0.09 0.9440.9540.9630.9710. That is correct! 0 point 0 4 1$$ 0 point 0 4 1 - correct Answer Explanation Correct answers:

• 0 point 0 4 1 $0.041$

The p - value is the probability of an observed value of z=1.74 or greater if the null hypothesis is true, because this hypothesis test is right-tailed. This probability is equal to the area under the Standard Normal curve to the right of z =1.74.

A standard normal curve with two points labeled on the horizontal axis. The mean is labeled at 0.00 and an observed value of 1.74 is labeled. The area under the curve and to the right of the observed value is shaded. Using the Standard Normal Table, we can see that the p - value is equal to 0.959, which is the area to the left of z =1.74. (Standard Normal Tables give areas to the left.) So, the p - value we're looking for is p =1−0.959=0.041. Question 6 1/1 points Kenneth, a competitor in cup stacking, claims that his average stacking time is 8.2 seconds. During a practice session, Kenneth has a sample stacking time mean of 7.8 seconds based on 11 trials. At the 4% significance level, does the data provide sufficient evidence to conclude that Kenneth's mean stacking time is less than 8.2 seconds? Accept or reject the hypothesis given the sample data below.

• H 0: μ =8.2 seconds; Ha : μ <8.2 seconds
• α =0.04 (significance level)
• z 0=−1.
• p =0. That is correct!

Do not reject the null hypothesis because the p - value 0.0401 is greater than the significance level α =0.04. Reject the null hypothesis because the p - value 0.0401 is greater than the significance level α =0.04. Reject the null hypothesis because the value of z is negative. Reject the null hypothesis because |−1.75|>0.04. Do not reject the null hypothesis because |−1.75|>0.04. Answer Explanation Correct answer: Do not reject the null hypothesis because the p - value 0.0401 is greater than the significance level α =0.04. In making the decision to reject or not reject H 0, if α > p - value, reject H 0 because the results of the sample data are significant. There is sufficient evidence to conclude that H 0 is an incorrect belief and that the alternative hypothesis, Ha , may be correct. If α ≤ p - value, do not reject H 0. The results of the sample data are not significant, so there is not sufficient evidence to conclude that the alternative hypothesis, Ha , may be correct. In this case, α =0.04 is less than or equal to p =0.0401, so the decision is to not reject the null hypothesis.

QUESTION 7 1/1 POINTS A recent study suggested that 81% of senior citizens take at least one prescription medication. Amelia is a nurse at a large hospital who would like to know whether the percentage is the same for senior citizen patients who go to her hospital. She randomly selects 59 senior citizens patients who were treated at the hospital and finds that 49 of them take at least one prescription medication. What are the null and alternative hypotheses for this hypothesis test? That is correct!

{H0:p=0.81Ha:p>0. {H0:p≠0.81Ha:p=0. {H0:p=0.81Ha:p<0. {H0:p=0.81Ha:p≠0. Answer Explanation Correct answer: {H0:p=0.81Ha:p≠0. First verify whether all of the conditions have been met. Let p be the population proportion for the senior citizen patients treated at Amelia's hospital who take at least one prescription medication.

1. Since there are two independent outcomes for each trial, the proportion follows a binomial model.
2. The question states that the sample was collected randomly.
3. The expected number of successes, np=47.79, and the expected number of failures, nq=n(1−p)=11.21, are both greater than or equal to 5. Since Amelia is testing whether the proportion is the same, the null hypothesis is that p is equal to 0.81 and the alternative hypothesis is that p is not equal to 0.81. The null and alternative hypotheses are shown below. {H0:p=0.81Ha:p≠0. QUESTION 8 1/1 POINTS

A researcher claims that the proportion of cars with manual transmission is less than 10%. To test this claim, a survey checked 1000 randomly selected cars. Of those cars, 95 had a manual transmission. The following is the setup for the hypothesis test: {H0:p=0.10Ha:p<0. Find the test statistic for this hypothesis test for a proportion. Round your answer to 2 decimal places. That is correct! $$Test_Statistic=−0. Answer Explanation Correct answers:

• $\text{Test_Statistic}=-0.53$Test_Statistic=−0. The proportion of successes is p^=951000=0.095. The test statistic is calculated as follows: z=p^−p0p0⋅(1−p0)n−−−−−−√ z=0.095−0.100.10⋅(1−0.10)1000−−−−−−−−√ z≈−0. QUESTION 9 1/1 POINTS A medical researcher claims that the proportion of people taking a certain medication that develop serious side effects is 12%. To test this claim, a random sample of 900 people taking the medication is taken and it is determined that 93 people have experienced serious side effects.. The following is the setup for this hypothesis test: H 0: p = 0.

Ha : p ≠ 0. Find the p-value for this hypothesis test for a proportion and round your answer to 3 decimal places. The following table can be utilized which provides areas under the Standard Normal Curve: That is correct! $$P-value=0. Answer Explanation Correct answers:

• $\text{P-value=}0.124$P-value=0. Here are the steps needed to calculate the p-value for a hypothesis test for a proportion:

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.

1.8 0.036 0.035 0.034 0.034 0.033 0.032 0.031 0.031 0.030 0.

1.7 0.045 0.044 0.043 0.042 0.041 0.040 0.039 0.038 0.038 0.

1.6 0.055 0.054 0.053 0.052 0.051 0.049 0.048 0.047 0.046 0.

1.5 0.067 0.066 0.064 0.063 0.062 0.061 0.059 0.058 0.057 0.

1.4 0.081 0.079 0.078 0.076 0.075 0.074 0.072 0.071 0.069 0.

1. Determine if the hypothesis test is left tailed, right tailed, or two tailed.
2. Compute the value of the test statistic.
3. If the hypothesis test is left tailed, the p-value will be the area under the standard normal curve to the left of the test statistic z 0 If the test is right tailed, the p-value will be the area under the standard normal curve to the right of the test statistic z 0 If the test is two tailed, the p-value will be the area to the left of −| z 0| plus the area to the right of | z 0| under the standard normal curve For this example, the test is a two tailed test and the test statistic, rounding to two decimal places, is z=0.1033−0.120.12(1−0.12)900−−−−−−−−−−−−√≈−1.54. Thus the p-value is the area under the Standard Normal curve to the left of a z-score of - 1.54, plus the area under the Standard Normal curve to the right of a z-score of 1.54. From a lookup table of the area under the Standard Normal curve, the corresponding area is then 2(0.062) = 0.124. QUESTION 10 1/1 POINTS

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.

1.8 0.036 0.035 0.034 0.034 0.033 0.032 0.031 0.031 0.030 0.

1.7 0.045 0.044 0.043 0.042 0.041 0.040 0.039 0.038 0.038 0.

1.6 0.055 0.054 0.053 0.052 0.051 0.049 0.048 0.047 0.046 0.

1.5 0.067 0.066 0.064 0.063 0.062 0.061 0.059 0.058 0.057 0.

1.4 0.081 0.079 0.078 0.076 0.075 0.074 0.072 0.071 0.069 0.

An economist claims that the proportion of people who plan to purchase a fully electric vehicle as their next car is greater than 65%. To test this claim, a random sample of 750 people are asked if they plan to purchase a fully electric vehicle as their next car Of these 750 people, 513 indicate that they do plan to purchase an electric vehicle. The following is the setup for this hypothesis test: H0:p=0. Ha:p>0. In this example, the p-value was determined to be 0.026. Come to a conclusion and interpret the results for this hypothesis test for a proportion (use a significance level of 5%.) That is correct! The decision is to reject the Null Hypothesis. The conclusion is that there is enough evidence to support the claim. The decision is to fail to reject the Null Hypothesis. The conclusion is that there is not enough evidence to support the claim. Answer Explanation Correct answer: The decision is to reject the Null Hypothesis. The conclusion is that there is enough evidence to support the claim. To come to a conclusion and interpret the results for a hypothesis test for proportion using the P- Value Approach, the first step is to compare the p-value from the sample data with the level of significance.

The decision criteria is then as follows: If the p-value is less than or equal to the given significance level, then the null hypothesis should be rejected. So, if p≤α, reject H0; otherwise fail to reject H0. When we have made a decision about the null hypothesis, it is important to write a thoughtful conclusion about the hypotheses in terms of the given problem's scenario. Assuming the claim is the null hypothesis, the conclusion is then one of the following:

• if the decision is to reject the null hypothesis, then the conclusion is that there is enough evidence to reject the claim.
• if the decision is to fail to reject the null hypothesis, then the conclusion is that there is not enough evidence to reject the claim. Assuming the claim is the alternative hypothesis, the conclusion is then one of the following:
• if the decision is to reject the null hypothesis, then the conclusion is that there is enough evidence to support the claim.
• if the decision is to fail to reject the null hypothesis, then the conclusion is that there is not enough evidence to support the claim. In this example, the p-value = 0.026. We then compare the p-value to the level of significance to come to a conclusion for the hypothesis test. In this example, the p-value is less than the level of significance which is 0.05. Since the p-value is greater than the level of significance, the conclusion is to reject the null hypothesis. QUESTION 11 1/1 POINTS Becky's statistics teacher was teaching the class how to perform the z-test for a proportion. Becky was bored because she had already mastered the test, so she decided to see if the coin she had in her pocket would come up heads or tails in a truly random fashion when flipped. She discretely flipped the coin 30 times and got heads 18 times. Becky conducts a one-proportion hypothesis test at the 5% significance level, to test whether the true proportion of heads is different from 50%.

Which answer choice shows the correct null and alternative hypotheses for this test? That is correct! H0:p=0.6; Ha:p>0.6, which is a right-tailed test. H0:p=0.5; Ha:p<0.5, which is a left-tailed test. H0:p=0.6; Ha:p≠0.6, which is a two-tailed test. H0:p=0.5; Ha:p≠0.5, which is a two-tailed test. Answer Explanation Correct answer: H0:p=0.5; Ha:p≠0.5, which is a two-tailed test. The null hypothesis should be true proportion: H0:p=0.5. Becky wants to know if the true proportion of heads is different from 0.5. This means that we just want to test if the proportion is not 0.5. So, the alternative hypothesis is Ha:p≠0.5, which is a two-tailed test. QUESTION 12 1/1 POINTS John owns a computer repair service. For each computer, he charges $50 plus $45 per hour of work. A linear equation that expresses the total amount of money John earns per computer

is y=50+45x. What are the independent and dependent variables? What is the y-intercept and the slope? That is correct! The independent variable (x) is the amount of time John fixes a computer. The dependent variable (y) is the amount, in dollars, John earns for a computer. John charges a one-time fee of $50 (this is when x=0), so the y-intercept is 50. John earns $45 for each hour he works, so the slope is 45. The independent variable (x) is the amount, in dollars, John earns for a computer. The dependent variable (y) is the amount of time John fixes a computer. John charges a one-time fee of $45 (this is when x=0), so the y-intercept is 45. John earns $50 for each hour he works, so the slope is 50. The independent variable (x) is the amount, in dollars, John earns for a computer. The dependent variable (y) is the amount of time John fixes a computer. John charges a one-time fee of $50 (this is when x=0), so the y-intercept is 50. John earns $45 for each hour he works, so the slope is 45. The independent variable (x) is the amount of time John fixes a computer. The dependent variable (y) is the amount, in dollars, John earns for a computer. John charges a one-time fee of $45 (this is when x=0), so the y-intercept is 45. John earns $50 for each hour he works, so the slope is 50.

Answer Explanation Correct answer: The independent variable (x) is the amount of time John fixes a computer. The dependent variable (y) is the amount, in dollars, John earns for a computer. John charges a one-time fee of $50 (this is when x=0), so the y-intercept is 50. John earns $45 for each hour he works, so the slope is 45. The independent variable (x) is the amount of time John fixes a computer because it is the value that changes. He may work different amounts per computer, and his earnings are dependent on how many hours he works. This is why the amount, in dollars, John earns for a computer is the dependent variable (y). The y-intercept is 50 (b=50). This is his one-time fee. The slope is 45 (a=45). This is the increase for each hour he works. QUESTION 13 1/1 POINTS Ariana keeps track of the amount of time she studies and the score she gets on her quizzes. The data are shown in the table below. Which of the scatter plots below accurately records the data? Hours studying Quiz score 1 5 2 5 3 7 4 9 5 9

That is correct! A scatterplot has a horizontal axis labeled Hours studying from 0 to 6 in increments of 1 and a vertical axis labeled Quiz score from 0 to 10 in increments of 2. The following points are plotted: left-parenthesis 1 comma 5 right-parentheses; left-parenthesis 2 comma 5 right-parentheses; left- parenthesis 3 comma 7 right-parentheses; left-parenthesis 4 comma 9 right-parentheses; left- parenthesis 5 comma 9 right-parentheses. All values are approximate.

A scatterplot has a horizontal axis labeled Hours studying from 0 to 10 in increments of 2 and a vertical axis labeled Quiz score from 0 to 6 in increments of 1. The following points are plotted: left-parenthesis 5 comma 1 right-parentheses; left-parenthesis 5 comma 2 right-parentheses; left- parenthesis 7 comma 3 right-parentheses; left-parenthesis 9 comma 4 right-parentheses; left- parenthesis 9 comma 5 right-parentheses. All values are approximate.

A scatterplot has a horizontal axis labeled Hours studying from 0 to 6 in increments of 1 and a vertical axis labeled Quiz score from 0 to 9 in increments of 1. The following points are plotted: left-parenthesis 1 comma 5 right-parentheses; left-parenthesis 2 comma 5 right-parentheses; left- parenthesis 3 comma 7 right-parentheses; left-parenthesis 4 comma 8 right-parentheses; left- parenthesis 5 comma 8 right-parentheses.

A scatterplot has a horizontal axis labeled Hours studying from 0 to 6 in increments of 1 and a vertical axis labeled Quiz score from 0 to 12 in increments of 2. The following points are plotted: left-parenthesis 1 comma 5 right-parentheses; left-parenthesis 2 comma 5 right-parentheses; left- parenthesis 3 comma 8 right-parentheses; left-parenthesis 4 comma 8 right-parentheses; left- parenthesis 5 comma 11 right-parentheses. All values are approximate. Answer Explanation Correct answer:

A scatterplot has a horizontal axis labeled Hours studying from 0 to 6 in increments of 1 and a vertical axis labeled Quiz score from 0 to 10 in increments of 2. The following points are plotted: left-parenthesis 1 comma 5 right-parentheses; left-parenthesis 2 comma 5 right-parentheses; left- parenthesis 3 comma 7 right-parentheses; left-parenthesis 4 comma 9 right-parentheses; left- parenthesis 5 comma 9 right-parentheses. All values are approximate. The values for hours studying correspond to x-values, and the values for quiz score correspond to y-values. Each row of the table of data corresponds to a point (x,y) plotted in the scatter plot. For example, the first row, 1,5, corresponds to the point (1,5). Doing this for every row in the table, we find the scatter plot should have points (1,5), (2,5), (3,7), (4,9), and (5,9). QUESTION 14 1/1 POINTS Data is collected on the relationship between time spent playing video games and time spent with family. The data is shown in the table and the line of best fit for the data is y^=−0.27x+57.5. Assume the line of best fit is significant and there is a strong linear relationship between the variables. Video Games (Minutes) 306090120 Time with Family (Minutes) 50403525 According to the line of best fit, the predicted number of minutes spent with family for someone who spent 95 minutes playing video games is 31.85. Is it reasonable to use this line of best fit to make the above prediction?

That is correct! The estimate, a predicted time of 31.85 minutes, is unreliable but reasonable. The estimate, a predicted time of 31.85 minutes, is both unreliable and unreasonable. The estimate, a predicted time of 31.85 minutes, is both reliable and reasonable. The estimate, a predicted time of 31.85 minutes, is reliable but unreasonable. Answer Explanation Correct answer: The estimate, a predicted time of 31.85 minutes, is both reliable and reasonable. The data in the table only includes video game times between 30 and 120 minutes, so the line of best fit gives reasonable predictions for values of x between 30 and 120. Since 95 is between these values, the estimate is both reliable and reasonable. QUESTION 15 0/1 POINTS Which of the following are feasible equations of a least squares regression line for the annual population change of a small country from the year 2000 to the year 2015? Select all that apply.

That's not right.

yˆ=38,000+2500x

yˆ=38,000−3500x

yˆ=−38,000+2500x

yˆ=38,000−1500x

Answer Explanation Correct answer: yˆ=38,000+2500x yˆ=38,000−1500x Population change can be positive or negative, and it can increase or decrease. Based on the given information, there are no practical limits to population change, although there are limits such as the decrease in population cannot exceed the current population (as it would leave a

negative number of people in the country), or the increase in population could be limited by other real-world factors (such as lack of space or legal immigration limits). Your answer: yˆ=38,000+2500x QUESTION 17 1/1 POINTS An amateur astronomer is researching statistical properties of known stars using a variety of databases. They collect the color index, or B−V index, and distance (in light years) from Earth for 30 stars. The color index of a star is the difference in the light absorption measured from the star using two different light filters (a B and a V filter). This then allows the scientist to know the star's temperature and a negative value means a hot blue star. A light year is the distance light can travel in 1 year, which is approximately 5.9 trillion miles. The data is provided below. Use Excel to calculate the correlation coefficient r between the two data sets, rounding to two decimal places. index Distance (ly) 1.1 1380 0.4 556 1.0 771 0.5 304 1.4 532