Question
A biologist gathered data on the length, in millimeters ( \(\mathrm{mm}\) ), and the mass, in grams ( \(\mathrm{g}\) ), for 11 bullfrogs. The data are shown in Plot 1.
(a) Based on the scatterplot, describe the relationship between mass and length, in context.
From the data, the biologist calculated the least-squares regression line for predicting mass from length. The least-squares regression line is shown in Plot 2.
(b) Identify and interpret the slope of the least-squares regression line in context.
(c) Interpret the coefficient of determination of the least-squares regression line, \(r^2 \approx 0.819\), in context.
(d) From Plot 2, consider the residuals of the 11 bullfrogs.
(i) Based on the plot, approximately what is the length and mass of the bullfrog with the largest absolute value residual?
(ii) Does the least-squares regression line overestimate or underestimate the mass of the bullfrog identified in part (d-i)? Explain your answer.
▶️Answer/Explanation
Ans:
(a) There appears to be a moderately strong linear relationship between mass and length for the 11 bullfrogs. There are no obvious outliers.
(b) The predicted mass in grams of a bullfrog is expected to increase by 6.086 for every \(1 \mathrm{~mm}\) increase in the length of a bullfrog.
(c)\(81.9 \%\) of the variation in the mass \((g)\) of a bull frog is accounted for by the least squares regression line on length \((\mathrm{mm})\) of a bullfrog.
(d)(i)The bullfrog with the largest absolute value residual has a length of approximately \(162 \mathrm{~mm}\) and a mass of approximately \(355 \mathrm{~g}\).
(ii) The least squares regression line would overestimate the mass of the bullfrog because the bullfrog has a negative residual with the observed value being less than the expected.
Question
A dermatologist will conduct an experiment to investigate the effectiveness of a new drug to treat acne. The dermatologist has recruited 36 pairs of identical twins. Each person in the experiment has acne and each person in the experiment will receive either the new drug or a placebo. After each person in the experiment uses either the new drug or the placebo for 2 weeks, the dermatologist will evaluate the improvement in acne severity for each person on a scale from 0 (no improvement) to 100 (complete cure).
(a) Identify the treatments, experimental units, and response variable of the experiment.
- Treatments:
- Experimental units:
- Response variable:
Each twin in the experiment has a severity of acne similar to that of the other twin. However, the severity of acne differs from one twin pair to another.
(b) For the dermatologist’s experiment, describe a statistical advantage of using a matched-pairs design where twins are paired rather than using a completely randomized design.
(c) For the dermatologist’s experiment, describe how the treatments can be randomly assigned to people using a matched-pairs design in which twins are paired.
▶️Answer/Explanation
Ans:
- Treatments: the type of drive received (new or place bol)
- Experimental units: The people participating in this expriment
- Response variable: Level of improvement after two weeks
A matches pairs design is statistically advantageous in this experiment become it rescues the effect of initial ache severity os d Confounding Variable. Since each twin in a pair has a simile dene severity to theofler, it is more effective to determine the drug’s effectiveness when composing between twins, is opposed to a randomized design where different ache Severities mad hake it harder to determine how r. much of an effect the drug octudily had.
For each pair of identical twin flip a fair coin to determine which twin gets the experiments l treatment. Whichever twin does not get the experimental treatment, gets the placebo. instead.
Question
A machine at a manufacturing company is programmed to fill shampoo bottles such that the amount of shampoo in each bottle is normally distributed with mean 0.60 liter and standard deviation 0.04 liter. Let the random variable \(A\) represent the amount of shampoo, in liters, that is inserted into a bottle by the filling machine.
(a) A bottle is considered to be underfilled if it has less than 0.50 liter of shampoo. Determine the probability that a randomly selected bottle of shampoo will be underfilled. Show your work.
After the bottles are filled, they are placed in boxes of 10 bottles per box. After the bottles are placed in the boxes, several boxes are placed in a crate for shipping to a beauty supply warehouse. The manufacturing company’s contract with the beauty supply warehouse states that one box will be randomly selected from a crate. If 2 or more bottles in the selected box are underfilled, the entire crate will be rejected and sent back to the manufacturing company.
(b) The beauty supply warehouse manager is interested in the probability that a crate shipped to the warehouse will be rejected. Assume that the amounts of shampoo in the bottles are independent of each other.
(i) Define the random variable of interest for the warehouse manager and state how the random variable is distributed.
(ii) Determine the probability that a crate will be rejected by the warehouse manager. Show your work.
To reduce the number of crates rejected by the beauty supply warehouse manager, the manufacturing company is considering adjusting the programming of the filling machine so that the amount of shampoo in each bottle is normally distributed with mean 0.56 liter and standard deviation 0.03 liter.
(c) Would you recommend that the manufacturing company use the original programming of the filling machine or the adjusted programming of the filling machine? Provide a statistical justification for your choice.
▶️Answer/Explanation
Ans:
$
\begin{aligned}
& z=\frac{x-\mu}{\sigma}=\frac{0.5-0.6}{0.04}=-2.5 \\
& P(A<0.5)=P(z<-2.5)=\text { norrealcdf }(-99,-0.5)=0.0060 \text { L. } 0.6 \text { shampoo }
\end{aligned}
$
There is about a \(0.62 \%\) chance a random bottle of shampoos will be undefiled.
(i) \(x=\) of bottles in 10 (one box) which will be underfilled \(\operatorname{Bino\mu }(n=10, p=.0062)\)
(ii) $
\begin{aligned}
P(x \geq 2)=1-P(x \leq 1)=1-P(x=0=1) & =1-b \text { inured }(n=10, p=0.0062, x=1)= \\
& =1-0.9983 \\
& =0.0017
\end{aligned}
$
There is about a \(0.13 \%\) chance
a crate will be rejected by the
Manager.
(c) \(z=\frac{x-\mu}{\sigma}=\frac{0.5-0.56}{0.03}=-2 \quad p(z<-2)=.023\)
The old programming is better. The old programming had a \(.62 \%\) chance for a bottle to be underfilled. However, with the adjusted prograrruing, there is a \(2.3 \%\) chance for a bottle to be undefiled. If they certain with the original programming, there is a lesser chance that a bottle will be under filled, and therefore a lesser chance of a rejected crate.
Question
A survey conducted by a national research center asked a random sample of 920 teenagers in the United States how often they use a video streaming service. From the sample, \(59 \%\) answered that they use a video streaming service every day.
(a) Construct and interpret a 95\% confidence interval for the proportion of all teenagers in the United States who would respond that they use a video streaming service every day.
(b) Based on the confidence interval in part (a), do the sample data provide convincing statistical evidence that the proportion of all teenagers in the United States who would respond that they use a video streaming service every day is not 0.5 ? Justify your answer.
▶️Answer/Explanation
Ans:
\(\begin{gathered}920 \cdot 0.59=542.8 \quad l f=920-1=919 \\ c=0.59 \pm t^\alpha \sqrt{\frac{p(1-\hat{p}}{n}} \\ =0.59 \pm 1.96 \sqrt{\frac{0.59(1-0.59)}{120}} \\ c=(0.562,0.618)\end{gathered}\)
(ii)Based on tho confidence interval of \((0,532,0,618)\), there is a risible chance that the proportion of all sens in tho US who would respond that they use a; videos sin caning service is not 0.5 . With the confidence interval we can chin that the true proportition of toes in the US whoilwoild respond that they use a video streaming service is within? the impanel \((0,562,0,618)\) with \(95 \%\) confilemes,
Question
Studies have shown that foods rich in compounds known as flavonoids help lower blood pressure. Researchers conducted a study to investigate whether there was a greater reduction in blood pressure for people who consumed dark chocolate, which contains flavonoids, than people who consumed white chocolate, which does not contain flavonoids. Twenty-five healthy adults agreed to participate in the study and add 3.5 ounces of chocolate to their daily diets. Of the 25 participants, 13 were randomly assigned to the dark chocolate group and the rest were assigned to the white chocolate group. All participants had their blood pressure recorded, in millimeters of mercury ( \(\mathrm{mmHg}\) ), before adding chocolate to their daily diets and again 30 days after adding chocolate to their daily diets.
The reduction in blood pressure (before minus after) for each of the participants in the two groups is shown in the dotplots below.
(a) Determine and compare the medians of the reduction in blood pressure for the two groups.
The researchers found the mean reduction in blood pressure for those who consumed dark chocolate is \(\bar{x}_{\text {dark }}=6.08 \mathrm{mmHg}\) and the mean reduction in blood pressure for those who consumed white chocolate is \(\bar{x}_{\text {white }}=0.42 \mathrm{mmHg}\).
(b) One researcher indicated that because the difference in sample means of \(5.66 \mathrm{mmHg}\) is greater than 0 there is convincing statistical evidence to conclude that the population mean blood pressure reduction for those who consume dark chocolate is greater than for those who consume white chocolate. Why might the researcher’s conclusion, based only on the difference in sample means of \(5.66 \mathrm{mmHg}\), not necessarily be true?
A simulation was conducted to investigate whether there is a greater reduction of blood pressure for those who consume dark chocolate than for those who consume white chocolate. The simulation was conducted under the assumption that no difference exists. The results of 120 trials of the simulation are shown in the following dotplot.
(c) Use the results of the simulation to determine whether the results from the 25 participants in the study provide convincing statistical evidence, at a 5 percent level of significance, that adding dark chocolate to a daily diet will result in a greater reduction in blood pressure, on average, than adding white chocolate to a daily diet. Justify your answer.
▶️Answer/Explanation
Ans:
(a) The median blood pressure reduckim is 1 for the doric Chocolate grove and 0 for the white Chocolate sue. The dart chocolate gran wis \(\partial\) higher median blood pressure reduction.
(b) you cannot conclude statistic significance based on sample means don. An appreciate hypothesis test must be conducted to decide whether or not “. lively the difference in Sample mean l was caused by random chance.
(b) You can not conclude statistics significance based on sample means alone. An appreciate hypothesis test must be conducted to decide whether or not o lively the difference in Sample mean l was caused by random chance.
Question
To compare success rates for treating allergies at two clinics that specialize in treating allergy sufferers, researchers selected random samples of patient records from the two clinics. The following table summarizes the data.
(a) (i) Complete the following table by recording the relative frequencies of successful and unsuccessful treatments at each clinic.
(ii) Based on the relative frequency table in part (a-i), which clinic is more successful in treating allergy sufferers? Justify your answer.
(b) Based on the design of the study, would a statistically significant result allow the researchers to conclude that receiving treatments at the clinic you selected in part (a-ii) causes a higher percentage of successful treatments than at the other clinic? Explain your answer.
▶️Answer/Explanation
Ans:
(a) Clinic a appears more successful in treating allergy sufferers, as demonstrated by their greater percentage of successful treatments. clinic A saw \(63.3 \%\) of treatments were successful, while clinic \(B\) only snows \(51.4 \%\) of treatments were successful
(b)No, researchers could not conclude that receiving treatments at clinic \(A\) would cause a nigher percentage of successful treatments because we are unaware if patients were randomly assigned to clinics. It is possible that far more severe patients go to clinic \(B\) and they have a correspondingly lower success rate. Although we are aware nat patient records were randomly selected from me clinic, we are unaware \(w\) he the the patients from clinic \(A\) are comparable to clinic \(B\) patents and therefore we cannot conclude a causal relationsnip between clinics and successes.