MATH 225 Week 8 Final Exam
Student Name
Chamberlain University
MATH-225 Statistical Reasoning for the Health Sciences
Prof. Name
Date
Question 1: Is it reasonable to use the line of best fit to predict time spent with family based on video game time?
Given the line of best fit y^=−0.27x+57.5\hat{y} = -0.27x + 57.5, the predicted number of minutes spent with family when someone plays video games for 95 minutes is 31.85 minutes. Since this value lies within the range of the original data and assumes a strong linear relationship, the prediction is considered both reliable and reasonable.
Correct Answer: The estimate, a predicted time of 31.85 minutes, is both reliable and reasonable.
Question 2: Which of the following are feasible equations for a least squares regression line?
To determine the appropriate equations for a regression line, we should look for equations with plausible slopes and intercepts that represent a potential trend over time. In this context, acceptable equations for population change include:
- y^=38,000+2500x\hat{y} = 38,000 + 2500x
- y^=38,000−3500x\hat{y} = 38,000 – 3500x
- y^=38,000−1500x\hat{y} = 38,000 – 1500x
The equation y^=−38,000+2500x\hat{y} = -38,000 + 2500x is not feasible due to the negative intercept.
Correct Answers:
- y^=38,000+2500x\hat{y} = 38,000 + 2500x
- y^=38,000−3500x\hat{y} = 38,000 – 3500x
- y^=38,000−1500x\hat{y} = 38,000 – 1500x
Question 3: Which group receives the placebo?
In the given experiment, the group that received the substance with no therapeutic effect is considered the placebo group. A placebo is used as a control to compare the effects of the actual medication.
Correct Answer: The group that received the drug with no therapeutic effect.
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Question 4: What are the correct null and alternative hypotheses for Becky’s coin flip experiment?
Becky flipped a coin 30 times and obtained heads 18 times. To determine if the proportion of heads is different from 0.5, she needs to perform a two-tailed hypothesis test. The null hypothesis should state that the proportion p=0.5p = 0.5, and the alternative hypothesis should state that p≠0.5p \neq 0.5, indicating a possible deviation in either direction from fairness.
Correct Answer: H0:p=0.5;Ha:p≠0.5H_0: p = 0.5; H_a: p \neq 0.5, which is a two-tailed test.
Question 5: Is the farmer’s study observational or experimental, and what is the controlled factor?
The farmer divided his land into four equal sections with similar soil conditions and planted the same crop, monitoring yields over four weeks. Since the farmer controlled land division and the planting conditions, this qualifies as an experiment. Among the listed options, the land was the manipulated factor.
Correct Answer: The study is an experiment. The controlled factor is the land.
Question 6: What is the minimum sample size required to estimate the mean within 2 inches with 95% confidence?
Given a population standard deviation of 3.7 inches and a 95% confidence level, the minimum required sample size nn can be calculated using the formula:
n=(z⋅σE)2n = \left( \frac{z \cdot \sigma}{E} \right)^2
Using a z-score of 1.96 for 95% confidence, the required sample size is approximately 14.
Correct Answer: 14 dog heights.
Question 7: What is the Type I error in the apartment affordability scenario?
In this scenario, the null hypothesis claims that Jacob earns enough to afford a luxury apartment. A Type I error occurs if Jacob wrongly rejects this hypothesis—believing he can’t afford the apartment when he actually can.
Correct Answer: Jacob thinks he does not earn enough money to afford the luxury apartment when, in fact, he does.
Question 8: Which normal distribution has the smallest standard deviation?
When comparing multiple normal curves on the same graph, the distribution with the narrowest bell shape has the smallest standard deviation. Distribution B appears most concentrated around the mean.
Correct Answer: B
Question 9: What is the level of measurement for the patient’s temperature data?
The doctor records the patient’s temperature every hour. Temperature data measured in degrees Fahrenheit is quantitative and can have meaningful intervals but no true zero, placing it on the interval level of measurement.
Correct Answer: Interval
Question 10: What is the median truck price?
Given the list of prices in thousands—20, 46, 19, 14, 42, 26, 33—when ordered, the median value (middle value) is 26.
Correct Answer: Median = 26 thousand dollars.
Question 11: What is the mode for the number of cards drawn until a queen appears?
The list of cards drawn is: 3, 12, 3, 11, 5, 5, 3, 10, 12. The number 3 appears most frequently, making it the mode.
Correct Answer: Mode = 3 cards.
Question 12: What does the histogram suggest about the data distribution?
Based on the histogram (not shown here but referred to in the question), if one tail appears longer than the other, it indicates skewness. If both sides are roughly balanced, it is symmetrical.
Correct Answer: The data are symmetric.
Question 13: Which hypothesis tests are left-tailed?
Left-tailed tests have alternative hypotheses indicating values less than the null hypothesis value (e.g., Ha:X<μH_a: X < \mu). Among the choices:
Correct Answer:
- H0:X≥19.7,Ha:X<19.7H_0: X \geq 19.7, H_a: X < 19.7
Question 14: What are the correct null and alternative hypotheses to determine if 53% of people prefer Product A?
To test whether 53% of people prefer Product A, the null hypothesis should state p=0.53p = 0.53. Since the question checks if the actual proportion differs, the alternative should be p≠0.53p \neq 0.53, a two-tailed test.
Correct Answer: H0:p=0.53;Ha:p≠0.53H_0: p = 0.53; H_a: p \neq 0.53
Question 15: Which hypothesis test is right-tailed?
Right-tailed tests have the alternative hypothesis indicating a value greater than the null (e.g., Ha:X>μH_a: X > \mu).
Correct Answer:
- H0:X≤7.4,Ha:X>7.4H_0: X \leq 7.4, H_a: X > 7.4
Question 16: Construct a frequency table for DVDs rented using four classes
Data provided:
15, 31, 28, 19, 14, 18, 28, 19, 10, 19, 10, 24, 14, 18, 24, 27, 10, 18, 16, 23
Here is the grouped frequency table:
| Lower Class Limit | Upper Class Limit | Frequency |
|---|---|---|
| 10 | 15 | 6 |
| 15 | 20 | 6 |
| 20 | 25 | 4 |
| 25 | 31 | 4 |
MATH 225 Week 8 Final Exam
Question 17
The bar graph below shows the number of boys and girls in different classes.
What is the number of boys in the class with the highest count?
According to the graph (not shown here), the class with the highest number of boys has 23 boys.
MATH 225 Week 8 Final Exam
| Class | Number of Boys | Number of Girls |
|---|---|---|
| Highest | 23 | (Not specified) |
Question 18
The histogram below displays the weights of rainbow trout (in pounds) caught by visitors at a lake on a Saturday afternoon. According to this histogram, which range of weights (in pounds) contains the lowest frequency?
The weight range with the lowest number of trout caught is greater than 12.5 pounds but less than 14.5 pounds.
| Weight Range (pounds) | Frequency |
|---|---|
| > 12.5 and < 14.5 | Lowest |
| Other ranges | Higher |
Question 19
The weight of a car can influence the mileage the car obtains. A random sample of 20 cars’ weights and mileage is collected. Use Excel to find the best fit linear regression equation, where weight is the explanatory variable. Round the slope and intercept to three decimal places.
The calculated regression equation is:
y^=−1.181x+71.374\hat{y} = -1.181x + 71.374
where xx is the car’s weight and y^\hat{y} is the predicted mileage.
| Parameter | Value |
|---|---|
| Slope (bb) | -1.181 |
| Intercept (aa) | 71.374 |
Question 20
The following frequency table summarizes a set of data. What is the five-number summary?
| Value | Frequency |
|---|---|
| 2 | 6 |
| 6 | 2 |
| 7 | 1 |
| 8 | 1 |
| 10 | 1 |
| 12 | 1 |
| 14 | 1 |
| 17 | 6 |
| 20 | 3 |
| 21 | 1 |
| 22 | 1 |
| 23 | 1 |
| 25 | 1 |
| 29 | 1 |
The five-number summary (Minimum, Q1, Median, Q3, Maximum) is:
| Summary | Value |
|---|---|
| Minimum | 2 |
| Q1 | 6 |
| Median | 17 |
| Q3 | 20 |
| Maximum | 29 |
Question 21
Which of the following frequency tables show a skewed data set? Select all answers that apply.
- Table 1:
| Value | Frequency |
|---|---|
| 5 | 1 |
| 6 | 2 |
| 7 | 10 |
| 8 | 11 |
| 9 | 17 |
- Table 2:
| Value | Frequency |
|---|---|
| 5 | 1 |
| 6 | 3 |
| 7 | 8 |
| 8 | 10 |
| 9 | 13 |
| 10 | 26 |
- Table 3:
| Value | Frequency |
|---|---|
| 12 | 1 |
| 13 | 1 |
| 14 | 3 |
| 15 | 6 |
| 16 | 23 |
| 17 | 29 |
Skewed datasets are generally identified by a long tail on one side. Based on this:
| Table | Skewed? |
|---|---|
| 1 | Yes |
| 2 | Yes |
| 3 | No (approximately symmetric) |
Question 22
Kenneth, a cup stacking competitor, claims his average stacking time is 8.2 seconds. During practice, his sample mean was 7.8 seconds from 11 trials. At the 4% significance level, does the data provide enough evidence to conclude his mean stacking time is less than 8.2 seconds?
Using the hypothesis test with:
H0:μ=8.2andHa:μ<8.2H_0: \mu = 8.2 \quad \text{and} \quad H_a: \mu < 8.2
and a calculated test statistic of −1.75-1.75, the conclusion is to fail to reject the null hypothesis because ∣−1.75∣|-1.75| is less than the critical value for 4% significance. Hence, there is insufficient evidence that Kenneth’s mean stacking time is less than 8.2 seconds.
Question 23
A nurse wants to test if the proportion of senior citizens who take at least one prescription medication in her hospital matches the suggested 81%. She randomly selects 59 patients, with 49 taking medication. What are the null and alternative hypotheses?
The hypotheses are:
H0:p=0.81andHa:p≠0.81H_0: p = 0.81 \quad \text{and} \quad H_a: p \neq 0.81
The p-value is 0.026, which is less than the 5% significance level, so the null hypothesis is rejected.
Conclusion: There is sufficient evidence to support the claim that the proportion differs from 81%.
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Question 24
A statistics professor recently reviewed final exam scores for her introductory statistics class. She found that the mean score out of 100 was 77, and the margin of error was 10. Based on this information, construct a confidence interval for the mean score on the final exam.
Answer:
The confidence interval for the mean score is calculated by adding and subtracting the margin of error from the sample mean. Therefore, the confidence interval is (67, 87).
Question 25
In a psychological study designed to test a drug’s effect on reducing anxiety, participants were divided into two groups: one group received the anxiety-reduction pill, and the other was given an inert pill. Which group received the placebo?
Answer:
The group that received the inert pill serves as the placebo group in this study.
Question 26
True or False: The more shoes a manufacturer produces, the more shoes they sell.
Answer:
False. Producing more shoes does not necessarily guarantee more sales due to factors like demand, market saturation, and consumer preferences.
Question 27
Among several box and whisker plots representing different data sets, which plot corresponds to the smallest standard deviation?
Answer:
Data set C has the smallest standard deviation, as indicated by the shortest interquartile range and least spread in the box plot.
Question 28
Given two normal distribution plots, A and B, which of the following statements are true? Select all that apply.
| Statement | True/False |
|---|---|
| A has the larger mean | True/False (dependent on the graph) |
| B has the larger mean | True/False |
| The means of A and B are equal | True/False |
| A has the larger standard deviation | True/False |
| B has the larger standard deviation | True/False |
| The standard deviations of A and B are equal | True/False |
Answer:
(Answers depend on specific graph data. Common interpretations: One distribution has a larger mean if its peak is shifted right; larger spread indicates larger standard deviation.)
Question 29
A poll during a basketball season’s final game surveyed fans’ preferences about the winning team. From urban areas, 216 out of 374 residents favored the defending champions, while in rural areas, 304 out of 466 did so. The probability that the observed difference happened by chance is 0.03. What does this imply?
Answer:
The results are statistically significant at the 0.05 significance level, indicating that the proportion of rural fans wanting the defending champions to win differs from that of urban fans.
Question 30
A chef claims that the average meatball weighs less than 4 ounces. To test this, she weighs 14 meatballs, finding a sample mean of 3.7 ounces and knowing the population standard deviation is 0.5 ounces. At a 10% significance level, what is the z-score for the hypothesis test where:
- H0: μ ≥ 4
- Ha: μ < 4
Answer:
Using the formula for the z-test statistic:
z=xˉ−μ0σ/n=3.7−40.5/14=−2.24z = \frac{\bar{x} – \mu_0}{\sigma / \sqrt{n}} = \frac{3.7 – 4}{0.5 / \sqrt{14}} = -2.24
Question 31
What is the p-value of a right-tailed hypothesis test with a test statistic z0=1.74z_0 = 1.74?
Answer:
From the standard normal table, the p-value corresponding to z=1.74z = 1.74 is approximately 0.041.
Question 32
Define the Type II error for the following hypothesis: A building inspector claims no more than 15% of structures were built without permits.
Answer:
A Type II error occurs if the inspector concludes no more than 15% were built without permits when in reality, more than 15% were built without permits.
Question 33
How would you describe the shape of a given histogram?
Answer:
The histogram is uniform, meaning the frequencies are approximately equal across the bins.
Question 34
An amateur astronomer collects data on the color index (B−V) and distance (light years) of 30 stars. Using Excel, calculate the correlation coefficient rr between the two variables.
| B-V Index | Distance (ly) |
|---|---|
| 1.1 | 1380 |
| 0.4 | 556 |
| … | … |
| 0.89 | 91.7 |
Answer:
The calculated correlation coefficient is r=0.18r = 0.18, indicating a weak positive relationship.
Question 35
Complete the contingency table and find the number of students who neither play sports nor play an instrument.
Answer:
The number of students who do not participate in sports or play an instrument is 34.
Question 36
A medical researcher claims that 12% of people taking a certain medication develop serious side effects. A sample of 900 shows 93 with side effects. The hypotheses are:
- H0: p=0.12p = 0.12
- Ha: p≠0.12p \neq 0.12
Calculate the p-value, rounded to three decimal places.
Answer:
Using the normal distribution table and sample statistics, the p-value is approximately 0.124.
Question 37
True or False: The closer the average daily crop harvest is to the peak, the higher the harvest quantity.
Answer:
False. The average daily harvest does not necessarily indicate proximity to peak harvest.
Question 38
Brayden tosses a coin 500 times, getting 416 heads. The probability of this happening by chance (assuming a fair coin) is less than 0.01. What does this significance level mean?
Answer:
At the 0.01 significance level, the result suggests the coin is likely not fair.
Question 39
A line graph shows the number of TVs in a house by square footage. Which of the following best describes the relationship?
Answer:
There is a steady increase in both square footage and the number of TVs.
Question 40
An economist claims more than 65% of people plan to buy a fully electric vehicle next. A sample of 750 people shows 513 agree, with p-value = 0.026 for:
- H0: p=0.65p = 0.65
- Ha: p>0.65p > 0.65
Interpret the result at a 5% significance level.
Answer:
Reject the null hypothesis. There is sufficient evidence to support the economist’s claim.
Question 41
Alice sells boxes of candy at baseball games. Her sales for the games are: 16, 14, 14, 21, 15. Find the mean number of boxes sold.
The mean is calculated as:
Mean=16+14+14+21+155=805=16\text{Mean} = \frac{16 + 14 + 14 + 21 + 15}{5} = \frac{80}{5} = 16
Question 42
John charges $50 plus $45 per hour for computer repair. The equation for total earnings is y=50+45xy = 50 + 45x. Identify the independent variable, dependent variable, y-intercept, and slope.
| Variable | Description |
|---|---|
| Independent variable xx | Hours spent fixing the computer |
| Dependent variable yy | Total amount earned in dollars |
| y-intercept | 50 (fixed base charge, when x=0x=0) |
| Slope | 45 (charge per hour of work) |
Question 43
Hugo’s typing speed is normally distributed with mean 59 and standard deviation 15. He types 57 words per minute in a test. What is the z-score and its interpretation?
The z-score is:
z=57−5915=−0.133z = \frac{57 – 59}{15} = -0.133
This means 57 is 0.133 standard deviations left of the mean.
Question 44
Ariana tracks study time and quiz scores shown in a table. Which scatter plot correctly displays the data?
Answer: The first scatter plot matches the data