Practice 1 Solutions

1. Calculate the z-score for the observed difference. Explain the statistical significance at the 5% level.

$z=\frac{3.30}{1.66}=1.988$

Because $1.988>1.96$, the new menu shows statistically significant results for the average check amount at the 5% level.

2. Calculate the expected additional yearly revenue (earnings) if the new menu were to be implemented.

$$\begin{gathered} 31.95–28.65=3.30 \\ \left(3.30\right)\left(6000\right)=19,800 \\ \left(19800\right)\left(12\right)=237,600 \end{gathered}$$

One customer check increases $3.30. The restaurant should expect a revenue increase of $237,600 over one year (12 months) if an average of 6,000 customers are served monthly.

3. Determine the annual cost for implementing 200 new menus at all locations. 

$$\begin{gathered} \left(7.50\right)\left(2\right)=15 \\ \left(15\right)\left(4\right)\left(200\right)=12,000 \end{gathered}$$

Each menu needs to be replaced every six months for an annual cost of $15 each. With 200 menus needed at each of the four locations, the total annual cost would be $12,000.

4. Based on the revenue and costs, should the restaurant group adopt the new menu? Explain.

$\$237,600–\$12,000=\$225,600$

The restaurant group should adopt new menus. After the cost of the menus is subtracted from the revenue, the restaurant is expected to earn an additional $225,600 annually.

5. What assumptions did you make to complete the cost analysis? What additional information would strengthen your recommendation?

Sample:
Assumptions: number of menus needed at each location, each check represents one customer, no other costs affect the decision about getting a new menu

A stronger recommendation would require the exact number of menus needed, as well as any other costs that would affect purchasing new menus.

6. Calculate the z-score. Explain the statistical significance at the 5% level. 

$$\begin{gathered} 81.5–79.8=1.7 \\ z=\frac{1.7}{5.86}=0.290 \end{gathered}$$

Because $0.290<1.96$, the brain games do not produce significant results (at any level because the z-score is so close to zero).

7. Press Release: “Brain games improve students’ test scores by a full letter grade! Sign up now to improve grades for a minimal cost.” Does the technology company have evidence to support this headline? Explain.
   

Sample: The press release is misleading because an increase of 1.7 percentage points is not a full letter grade. The results show improvement from a C+ to a B–, which is not a full letter grade (10% points).

8. Suppose you purchased a month of the brain games and had no change in your test scores. Write a short response requesting a full refund. Remember to use statistics to justify your reasoning.

$\left(3.75\right)\left(12\right)=45$

Sample: To whom it may concern, I am requesting a full $45 refund. Your brain game products promised to improve my test scores by a full letter grade, which would be a 10% increase. Because my test scores have shown zero change since I started using this product, a refund is required.

9. Calculate the z-score for the observed differences of both recovery and pain score. Explain the statistical significance at the 5% level.

$\begin{array}{lcl}\mathrm{Recovery} \mathrm{time}& & \mathrm{Pain} \mathrm{score}\\ 13.8–11.2=2.6& & 3.6–2.8=1.2\\ z=\frac{2.6}{1.12}=2.321& & z=\frac{1.2}{0.62}=1.935\end{array}$

Because $2.321>1.96$, the new technique shows statistically significant recovery times. While the pain scores are reduced with the new technique, the difference is not statistically significant at the 5% level.

10. From the patient’s perspective, evaluate the practical significance of both recovery time and pain score.

Sample:  Practically speaking, both results are significant because patients are able to recover more quickly with the new treatment and have reduced pain at the same time.

11. Suppose the office treats 200 shoulder patients annually, and three physical therapists would need to be certified if the new technique were implemented. Should the practice invest in the new technique? (Remember to consider statistics as well as practical outcomes.)

$\$15,000+3\left(\$2,000\right)=\$21,000$

Sample: The initial cost of the new technique will be $21,000. If recovery time is reduced by 2.6 weeks for most patients, it is possible that additional patients could be seen at the office. If more patients are seen, this could cover the extra cost of switching to the new technique. Patients will like a shorter recovery time because it will reduce their medical costs. 

12. Explain what might have resulted in the variables showing mixed results. What could this mean for the physical therapist when making decisions?

Sample: The results could be mixed because a pain score is the patient’s personal feeling about their pain, and each person has a different pain tolerance. On the other hand, the recovery time is tracking time. A physical therapist needs to consider that with the two variables, they can see that the treatment reduces time and pain. Both are critical to patient recovery.

13. Calculate the z-score for the observed differences. Explain the statistical significance. 

$$\begin{gathered} 7.7–6.8=0.9 \\ z=\frac{0.9}{0.41}=2.195 \end{gathered}$$

Because $2.195>1.96$, not using your phone for one hour before bed is statistically significant at the 5% level.

14. Is the treatment meaningful despite it falling short of the recommended 8–10 hours of sleep each night? Explain. 

Sample: The treatment is meaningful because it is both statistically and practically significant. Students reporting they “feel more alert” is important for learning, and it shows in their improved test scores. 

15. Would you personally try putting your phone away one hour before bed? Explain your decision and consider the challenges of following this rule.

Sample: I would consider putting my phone away before bed. Based on the data, I would have a better night’s sleep. It could be challenging if I use a study app on my phone before bed or if I need to set a phone alarm.