If there is one prayer that you should pray/sing every day and every hour, it is the
LORD's prayer (Our FATHER in Heaven prayer)
It is the most powerful prayer.
A pure heart, a clean mind, and a clear conscience is necessary for it.
- Samuel Dominic Chukwuemeka
For in GOD we live, and move, and have our being. - Acts 17:28
The Joy of a Teacher is the Success of his Students. - Samuel Chukwuemeka
For ACT Students
The ACT is a timed exam...60 questions for 60 minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you
solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no negative penalty for any wrong answer.
For NSC Students For the Questions:
Any space included in a number indicates a comma used to separate digits...separating multiples of three digits from behind.
Any comma included in a number indicates a decimal point. For the Solutions:
Decimals are used appropriately rather than commas
Commas are used to separate digits appropriately.
Solve all questions.
Show all work.
Please Note: Do not round intermediate calculations.
If you must round intermediate calculations, round them to at least four decimal places more than the number of decimals required for the final answer.
For example; if the final answer requires rounding to 3 decimal places, then round your intermediate calculations to at least 7 decimal places.
(1.) (a.) Interpret the statement: 99% confidence interval is: 4.1 < μ < 5.2
We are 99% confident that the interval from 4.1 to 5.2 actually contain the true value of the
population mean.
(2.) Determine whether the interpretation is valid or invalid.
A university is trying to accommodate its commuter students in its course scheduling.
In a survey, the school asked a sample of 75 randomly chosen students how long their daily commute is, in minutes.
The data showed a 99% confidence interval of (74.4, 86.8) for the mean commute time for students at the university.
If 100 more surveys are conducted (each using a sample with members chosen randomly and independently), it is expected that exactly 99 of them will each produce a 99% confidence interval that contains its sample mean.
The interpretation is invalid.
If 100 more students are chosen randomly to do the survey, it is guaranteed that all 100, not just 99 of them, will each produce a 99% confidence interval that contains the sample mean.
(3.)
(4.) Determine whether the interpretation is valid or invalid.
A tire manufacturer wants to examine the durability of its new product.
Company engineers randomly selected 175 new tires and measured the number of kilometers each can drive before blowing out.
The engineers found a 95% confidence interval of (18157.6, 18522.6) for the mean distance their new tires can drive before blowing out.
If 100 more samples are taken (with elements chosen randomly and independently), it is expected that exactly 100 of them will each produce a 95% confidence interval that contains its sample mean.
The interpretation is valid.
If 100 more samples are taken (with elements chosen randomly and independently), it is expected that exactly 100 of them will each produce a 95% confidence interval that contains its sample mean.
(5.)
(6.) Determine whether the interpretation is valid or invalid.
A manager at a Cereal company wants to confirm that there is an acceptable number of raisins in the company's family-sized boxes.
She counted the number of raisins in 100 randomly sampled boxes from the factory.
From the collected data, she computed a 90% confidence interval of (333.7, 357.1) for the mean number of raisins in family-sized boxes of the cereal.
If 100 more samples are taken (with elements chosen randomly and independently), it is expected that exactly 10 of them will each produce a 90% confidence interval that does not contain its sample mean.
The interpretation is invalid.
If 100 more samples are taken, it is guaranteed that all 100 of them will each produce a 90% confidence interval that contains its sample mean.
In other words, the number of 90% confidence intervals that do not contain the sample mean is guaranteed to be 0.
It is not expected to be 10.
(7.) A sample is taken from a population.
The size of the sample is 36
The mean of the sample is 123
The standard deviation of the population is 10
(a.) Construct a 95% confidence interval for the population mean.
(b.) Interpret your result.
$
n = 36 \\[3ex]
\bar{x} = 123 \\[3ex]
\sigma = 10 \\[3ex]
CL = 95\% = 0.95 \\[3ex]
\alpha = 1 - CL \\[3ex]
\alpha = 1 - 0.95 \\[3ex]
\alpha = 0.05 \\[3ex]
\dfrac{\alpha}{2} = \dfrac{0.05}{2} = 0.025 \\[5ex]
$
The population standard deviation was given
So, we have to use the $z$ distribution
$
z_{\dfrac{\alpha}{2}} = z_{0.025} = 1.9600 \\[5ex]
E = \dfrac{\sigma * z_{\dfrac{\alpha}{2}}}{\sqrt{n}} \\[7ex]
E = \dfrac{10 * 1.96}{\sqrt{36}} \\[5ex]
E = \dfrac{19.6}{6} \\[5ex]
E = 3.2\bar{6} \\[3ex]
E \approx 3.27 \\[3ex]
95\%\:\: CI = \bar{x} \pm E \\[3ex]
95\%\:\:CI = 123 \pm 3.27 \\[3ex]
95\%\:\:CI = 119.7\bar{3} \:\:and\:\: 126.2\bar{6} \\[3ex]
\therefore 95\%\:\:CI = 119.73 \lt \mu \lt 126.27 \\[3ex]
$
We are 95% confident that the population mean is within the margin of error of the sample mean
of 123.
We are 95% confident that the population mean is between 119.73 and 126.27
(8.) Determine whether the interpretation is valid or invalid.
A farmer owned an orange grove and wanted to know how productive his trees were.
He randomly selected 75 trees in the grove and kept track of how many oranges each tree produced during one season.
He found a 99% confidence interval of (459.8, 481.6) for the mean number of oranges produced by his trees that season.
There is a 99% chance that the mean number of produced by his trees that season is in the interval (459.8, 481.6).
The interpretation is invalid.
The population mean is the average number of oranges produced by all of the farmer's trees that season, not just the ones in the sample.
(9.) An infomercial claimed that a woman drove $6$ hours without oil, thanks to an engine treatment.
To determine the effectiveness of engine treatment, a company in the City of Truth or Consequences, New Mexico
tested engines in which they added the treatment to the motor oil, ran the engines, drained the oil, and then determined
the time until the engines seized.
The times (in minutes) are listed as shown:
$13.15$
$15.05$
$13.62$
$16.64$
$17.04$
$16.66$
$14.41$
$15.10$
$17.21$
$17.87$
$24.04$
$14.39$
(a.) Determine the point estimate of the mean amount of time.
(b.) What is the biggest flaw of using a point estimate to determine the population mean?
(c.) Construct a 95% confidence interval of the population mean.
(d.) Interpret your result.
Waiting Times, $x$
Frequencies, $f$
$f * x$
$x - \bar{x}$
$(x - \bar{x})^2$
$f(x - \bar{x})^2$
$13.15$
$1$
$13.15$
$-3.115$
$9.703225$
$9.703225$
$15.05$
$1$
$15.05$
$-1.215$
$1.476225$
$1.476225$
$13.62$
$1$
$13.62$
$-2.645$
$6.996025$
$6.996025$
$16.64$
$1$
$16.64$
$0.375$
$0.140625$
$0.140625$
$17.04$
$1$
$17.04$
$0.775$
$0.600625$
$0.600625$
$16.66$
$1$
$16.66$
$0.395$
$0.156025$
$0.156025$
$14.41$
$1$
$14.41$
$-1.855$
$3.441025$
$3.441025$
$15.10$
$1$
$15.10$
$-1.165$
$1.357225$
$1.357225$
$17.21$
$1$
$17.21$
$0.945$
$0.893025$
$0.893025$
$17.87$
$1$
$17.87$
$1.605$
$2.576025$
$2.576025$
$24.04$
$1$
$24.04$
$7.775$
$60.450625$
$60.450625$
$14.39$
$1$
$14.39$
$-1.875$
$3.515625$
$3.515625$
$\Sigma f = 12$
$\Sigma fx = 195.18$
$\Sigma f(x - \bar{x})^2 = 91.3063$
$
(a.) \\[3ex]
\bar{x} = \dfrac{\Sigma fx}{\Sigma f} \\[5ex]
\bar{x} = \dfrac{195.18}{12} \\[5ex]
\bar{x} = 16.265 \\[3ex]
$
The point estimate of the population mean is the sample mean
The point estimate of the population mean = $16.265$ minutes
(b.) The flaw in using the point estimate is that it gives only a single value estimate of
the population mean as compared to the interval estimate which gives a range of values.
$
\Sigma f - 1 = 12 - 1 = 11 \\[3ex]
s = \sqrt{\dfrac{\Sigma f(x - \bar{x})^2}{\Sigma f - 1}} \\[5ex]
s = \sqrt{\dfrac{91.3063}{11}} \\[5ex]
s = \sqrt{8.30057273} \\[3ex]
s = 2.88107146 \\[3ex]
CL = 95\% = 0.95 \\[3ex]
\alpha = 1 - CL \\[3ex]
\alpha = 1 - 0.95 \\[3ex]
\alpha = 0.05 \\[3ex]
\dfrac{\alpha}{2} = \dfrac{0.05}{2} = 0.025 \\[5ex]
$
The population standard deviation was not given
So, we have to use the $t$ distribution
$
df = n - 1 = 12 - 1 = 11 \\[3ex]
t_{\dfrac{\alpha}{2}} = t_{0.025} = 2.1009 \\[5ex]
E = \dfrac{s * t_{\dfrac{\alpha}{2}}}{\sqrt{n}} \\[7ex]
E = \dfrac{2.88107146 * 2.1009}{\sqrt{12}} \\[5ex]
E = \dfrac{6.05284303}{3.46410162} \\[5ex]
E = 1.74730527 \\[3ex]
95\%\:\: CI = \bar{x} \pm E \\[3ex]
95\%\:\:CI = 16.265 \pm 1.74730527 \\[3ex]
95\%\:\:CI = 14.5176947 \:\:and\:\: 18.0123053 \\[3ex]
\therefore 95\%\:\:CI = 14.52 \lt \mu \lt 18.01 \\[3ex]
$
We are 95% confident that the population mean is within the margin of error of the sample mean
of 16.265
We are 95% confident that the population mean is between 14.52 and 18.01
(10.) Determine whether the interpretation is valid or invalid.
A psychiatrist conducted a study about the effect of sleep deprivation on academic performance.
As part of the study, she surveyed 50 random students from a certain High School about their recent sleep patterns.
From the sample measurements, the psychiatrist calculated a 90% confidence interval of (45.4, 53.2) for the mean number of hours the high school students slept last week.
90% of the high school students slept between 45.4 and 53.2 hours last week.
The interpretation is invalid.
By chance, the original survey might have included only students that are heavy sleepers and sleep significantly more hours than what is average for the high school students.
(11.)
(12.) Determine whether the interpretation is valid or invalid.
Complaints about poor service led the Department of Motor Vehicles (DMV) of a certain State to investigate the wait times at a local branch.
During one week, DMV employees randomly asked 50 people leaving this branch how long they had to wait in line.
Using the responses, they found a 99% confidence interval of (39.3, 42.1) for the mean wait time at the DMV branch during that week (in minutes).
At most 1% of people leaving this branch during the week waited less than 39.3 or more than 42.1 minutes.
The interpretation is invalid.
By chance, the survey might have included people who got stuck in long lines and had to wait significantly more than what is average.
(13.)
(14.) Determine whether the interpretation is valid or invalid.
A store owner wants to check whether her competitor carries larger produce than she does.
She purchased 100 randomly selected kiwis from her competitor and weighed them.
The owner found a 99% confidence interval of (40.9, 41.9) for the mean weight of kiwis from her competitor (in grams).
If 100 more samples are taken (with elements chosen randomly and independently), it is expected that the mean weight of kiwis from her competitor would be in 99 of the computed 99% confidence intervals.
The interpretation is valid.
For 99% of random samples, the 99% confidence interval will contain the population mean.
This means that if 100 more random samples are taken, it is expected that the calculated interval for 99 of them will contain the population mean.
(15.)
(16.) Determine whether the interpretation is valid or invalid.
A man is convinced that the German novel he just finished reading has some of the longest sentences he has ever encountered.
To prove this to his friends, he randomly selected 50 sentences in the novel and noted the number of words in each.
He found a 95% confidence interval of (20.2, 24.2) for the mean number of words in sentences from the novel.
If 100 more samples are taken (with elements chosen randomly and independently), it is expected that for 95 of the samples, the sample mean will be between 20.2 and 24.2.
This is invalid.
By chance, the original sample might have included only elaborate sentences significantly longer than what is average for the novel.
(17.)
(18.) Determine whether the interpretation is valid or invalid.
A physical trainer is designing a fitness program for middle-aged men.
She asked a random sample of 175 men in their 40s to report how far they could jump in centimeters from a standstill.
The trainer computed a 99% confidence interval of (121.9, 127.1) for the mean jump distance for men in their 40s.
If the trainer takes another random sample, there is a 1% chance that the mean jump distance for men in their 40s will not be in the new sample's 99% confidence interval.
The interpretation is valid.
For 99% of random samples, the 99% confidence interval will contain the population mean.
So, for the other 1% of random samples, the 99% confidence interval will not contain the population mean.
This means that if the sample were repeated, there would be a 1% chance the confidence interval for that sample would not contain the population mean.
(19.)
(20.) Determine whether the interpretation is valid or invalid.
During a recent exhibit on the physics of football, volunteers at the local science museum conducted a survey of 750 randomly chosen visitors.
The survey included questions about attendance at sporting events.
The volunteers calculated a 90% confidence interval of (7.9, 8.1) for the mean number of sporting events museum visitors attended last year.
The average number of sporting events visitors in the survey went to last year is between 7.9 and 8.1.
The interpretation is valid.
The confidence interval (7.9, 8.1) contains the sample mean.
In other words, the average number of sporting events visitors in the survey went to last year is between 7.9 and 8.1.
(22.) Determine whether the interpretation is valid or invalid.
Game reviewers complained extensively about how a gaming company's recent game was too short.
To ensure this does not happen again, the gaming company gave a pre-release copy of its new game to 100 randomly selected gamers.
Each gamer was asked to note his or her play-through time.
The gaming company found a 90% confidence interval of (214.6, 242) for the mean play-through time for all gamers in minutes.
There is a 10% chance that the mean play-through time for all gamers (in minutes) is not in the interval (214.6, 242).
The interpretation is invalid.
The population mean is the average play-through time in minutes for all gamers, not just the ones in the sample.
(23.)
(24.) Determine whether the interpretation is valid or invalid.
An epidemiologist interested in how the common cold impacts health clinics.
She asked 275 randomly selected clinics across a certain State for their patient data from one specific month.
For each clinic, she looked at the number of patients who came in with a common cold that month.
She determined a 90% confidence interval of (43.5, 46.9) for the mean number of patients with a common cold who visited clinics in the State that month.
If she takes another random sample, there is a 90% chance that the mean number of patients with a common cold who visited clinics in the State that month will be in the new sample's 90% confidence interval.
The interpretation is valid.
For 90% of random samples, the 90% confidence interval will contain the population mean.
This means that if the sample were repeated, there would be a 90% chance the confidence interval for that sample would contain the population mean.
(25.)
(26.) Determine whether the interpretation is valid or invalid.
Executives at a certain public park would like to know what age group to target with their advertisements.
To begin, the executives collected the ages of 400 randomly selected attendees in one month.
Analysis of the study produced a 99% confidence interval of (13.2, 14.2) for the mean age of attendees at the park.
If the executives take more same-sized samples, then 99% of the attendees in each sample will be between 13.2 and 14.2 years old.
The interpretation is invalid.
By chance, the original sample might have included only attendees significantly older than what is average.
If this were the case, the confidence interval (13.2, 14.2) would exclude much of the population.
As a result, in most samples it would not be the case that 99% of sampled attended would be between 13.2 and 14.2 years old.
(27.)
(28.) Determine whether the interpretation is valid or invalid.
Botanists are concerned that rising temperatures are impeding the growth of trees in a certain National Forest.
To investigate, they measured the heights of a random sample of 50 mature birch trees to the nearest tenth of a meter.
They calculated a 95% confidence interval of (11.7, 13.9) for the mean height of mature birch trees in the National Forest.
If 100 more samples are taken (with elements chosen randomly and independently), it is expected that the mean height of mature birch trees in the forest would be in 95 of the computed 95% confidence intervals.
The interpretation is valid.
The population mean is the average height of all mature birch trees in the National Forest, not just the ones in the sample.
For 95% of random samples, the 95% confidence interval will contain the population mean.
This means that if 100 more random samples are taken, it is expected that the calculated interval for 95 of them will contain the population mean.
For ACT Students