Exst 2201 - Exam 2

What type of information do we really wish the science of statistics would give us so that we could make a good decision about an important matter?

Information about the population.

Why does a column of population data values contain more information about the population than a column of sample data values?

A column of population data values has more data values than a column of sample data values.

What are statistical methods of inference used for?

To get information about population data values.

Which one of the following choices does NOT indicate why knowing the value of the population mean is so important in the science of statistics?

Because the population mean was one of the first statistical methods developed.

In regard to normally distributed data values themselves, what is the importance of knowing the value of the population mean in the science of statistics?

Most of the data values are clustered close to the value of the mean.

In the common, real-world practice of statistics, is it ever possible to know the true value of the population mean?

All these other choices.

Why is the sample average important in the science of statistics?

It best estimates the value of the population mean from the sample data values.

What is the end result of the confidence interval method?

An interval estimate of the population mean.

What is the end result of the hypothesis test method?

A conclusion to a statistical test about a population mean.

Choose the two parts of Step A: Abstract below?
Calculate the test statistic.
Find the critical value.
Write the two hypotheses.
Get descriptive statistics.

Write the two hypotheses.
Get descriptive statistics.

Choose the two tools used in Step 1: Theorize below?
The Probability Calculator.
The Schematic z-Curve.
The Equation for sample average.
The z-Table.

The Schematic z-Curve.
The z-Table.

What is the primary task in Step 2: Analyze?

Choose the appropriate equation.

What is the end result of Step 3: Infer?

Statistical information about the value of a population mean.

Why does having more sample data values mean getting more information about the population data values?

Because each data value brings a bit more information about the population data values.

What type of statistics are used to learn about sample data values?

Descriptive statistics, because they describe the data values in a column of sample data values.

Please select the choice that is NOT a purpose of the Process of Abstraction?

Summarize the question text into a reasonable size.

The statement, 'On average, Americans spend around 4.5 hours per day on their phones.' has what interpretation in the process of abstraction?

That all Americans spend 4.5 hours.

What is the correct symbol for the significance level of a statistical study?

Alpha (α).

What two tools are used to go between the blanks on the Algebraic side of the Schematic Curve for Sample Averages?
The z-Table.
This z-Equation: z = (x̄ − μ) / (σ / √n).
This z-Equation: z = (x − μ) / σ.
The x-Table.

The z-Table.
This z-Equation: z = (x̄ − μ) / (σ / √n).

What actions occur in the Analyze step of the General Method to Solve Problems in Statistics?

The calculations for the method are chosen and performed.

What actions occur in the Infer step of the General Method to Solve Problems in Statistics?

Determining a population value from a sample values.

Two common statistical estimators of the population mean are the sample average and the confidence interval. Which one of these statistics is more useful to give information about the value of a population mean?

Confidence interval, as it shows how far the mean might be from the average.

Why is the confidence interval method the best method to get an interval estimate of the population mean?

It uses the most information from the distribution of the data values.

What is a major weakness in the method of a confidence interval?

A confidence interval does not give a degree of reasonableness for any value to be the population mean.

The logic of a confidence interval comes from reversing the logic of the sampling distribution of the sample average. Thus, the schematic curve for a confidence interval is centered over which statistic for location?

Sample average (x̄).

What is the name of the spread in a confidence interval?

The margin of error.

Select two choices below that make up the two parts of a margin of error in a confidence interval?
The level of confidence in the confidence interval method.
The level of predictiveness in the confidence interval method.
The standard error of the sample average.
The standard deviation of the data values.

The level of confidence in the confidence interval method.
The standard error of the sample average

The schematic curve for a confidence interval divides the curve into several regions. Which region will always contain the population mean?

Confidence region.

The size of the confidence region is given by the size of the confidence level. What is a simple meaning of the confidence level?

How many times out of 100 the method will give a correct answer.

What points on the real number line separate the confidence region from the significance regions?

The critical values.

What is the size of the significance region?

Alpha (α).

Is it guaranteed that confidence interval method will give an interval containing the population mean every time?

No, the method will give an incorrect interval alpha (α) percent of the time.

In a confidence interval, what information does the confidence level give?

The proportion of times the confidence interval includes the population mean.

In the science of statistics, why is the method of confidence intervals needed to get information about the population mean?

As the population mean cannot be determined exactly from probabilistic sample data values.

The statistical logic for a confidence interval is based on reversing already known logic for another statistic. What is this other logic?

The logic for the sampling distribution of the sample average.

Is the interval shown below properly denotated for a confidence interval?
(115,97)

No, values in the parentheses must go from lowest to highest.

Looking in a schematic curve from the point of view of probability, what type of situation is a confidence interval?

A middle body area.

The lower and upper confidence limits for a confidence interval can be seen in a schematic curve. The two confidence limits are just the x-values of a middle body area in a schematic curve.

True.

What is the main strength of a confidence interval?

Gives a range of likely values for the population mean.

What is the major weakness of a confidence interval?

Every value inside the interval is equally likely to be the population mean.

What is the correct equation for a confidence interval with t-values?

CI% = x̄ ± t(α/2,n−1)(s/√n)

Consider the size of the significance region, what is its appropriate symbol and how is it calculated?

α = 1 − CL

Will every confidence interval contain the value of the population mean?

No, only the confidence level proportion of the time.

What is the critical value for a 95% confidence interval with 22 data values and the sample standard deviation?

Critical value = 2.080

What is the critical value for a 90% confidence interval with 31 data values and the sample standard deviation?

Critical value = 1.697

What is the critical value for a 99% confidence interval with 10 data values and the sample standard deviation?

Critical value = 3.250

In the confidence interval equation shown below, what part of the equation is the margin of error?
CI% = x̄ ± t(α/2,n−1)(s/√n)

ME = t(α/2,n−1)(s/√n)

If the degrees of freedom gives the number of units of information, what does the margin of error give?

Half the width of a confidence interval.

When the confidence level is increased, how does the width of the confidence interval change?

Widens, as the middle body area gets larger

When the number of data values is increased, how does the width of the confidence interval change?

Narrows, as the standard error of the sample average gets smaller.

Two pieces of information must be known to use the t-distribution to find the probability of an event. What are those two pieces of information?

Sample standard deviation / Degrees of freedom.

What is the value of the sample average / value of the margin of error in the confidence interval shown below?
(96, 118)

Values = 107 / 11.

Is the following a true statement? 'A confidence interval is used to infer the value of the sample average?

No, as it is used to find the value of the population mean.

Is the following a true statement? 'The population mean must be known to calculate a confidence interval?

No, as finding the value of the population mean is the goal.

What is the meaning of the values inside a properly made confidence interval?

Any of these values could be a likely value for the population mean.

A local college instructor gave a statistics exam to his class of 31 students and calculated a 95% confidence interval of (77.1, 82.9) for the scores. A competing college knew that their students scored an average of 85 on this same test. How well did the local college do in comparison to the competing college?

Not well, as 85 is higher than the upper confidence limit.

31 students in a local statistics class took an exam where the average score was 80, and the standard deviation score was 10. Students in a competing statistics class took the same exam and had an average score of 85. Did the competing students score higher than the population mean of the local students? Make a 95% confidence interval to find out.

Yes, as CI = (76.3, 83.7).

The ages of 21 people in an elder care home had an average age of 85 years and a standard deviation age of 8 years. Could the population mean age of people in an elder care home be 87 years? Make a 90% confidence interval to find out.

Yes, as 87 is inside of CI = (81.989, 88.011).

In 2015 on the National Assessment of Educational Progress (NAEP) science scale, the following results were obtained.
Score for 15 underage students: average = 125, standard deviation = 5.81.
Score for 15 over age students: average = 136, standard deviation = 3.87.
Both data shapes were normal. Please make two 98% confidence intervals to find out if it is reasonable that these two groups of students have the same population mean score.

No, as the two confidence intervals, (121.1, 128.9) and (133.4, 138.6) do not overlap.

An auto mechanic wants to estimate the cost of repairing a car after a 25 mph collision. He looked up 24 collisions and found that the average repair cost was $11,000 with a standard deviation cost of $2,500. Please find the middle value / spread value of a 98% confidence interval for the population mean repair cost.

Values = $11,000 / $1,276.

Last year a sample of 35 women who were victims of purse snatching had an average age of 34.25 years, with a standard deviation age of 10 years. One researcher wondered if the population mean age of all victims of purse snatching could be less than 30 years. Make a 96% confidence interval to find out?

No, as the confidence interval for age = (30.64, 37.86).

The height of 14 students in a college statistics course was measured and found to have an average height of 68 inches, and a standard deviation height of 2 inches. What is the 90% confidence interval for the mean height of all students at the college?

CI = (67.0, 68.9).

The height of 27 students in a college statistics course was measured and found to have a 90% confidence interval height of (65.9, 69.9). Is it possible that the mean height of all the students at this college 5 feet 6 inches (66 inches) or higher?

Yes, as 66 inches is inside the confidence interval.

The confidence interval shown below was calculated from a sample average of 100 and a margin of error of 15. Is 100 a more likely value for the population mean than 90?
(85, 115)

No, as all values inside the confidence interval are equally likely.

A fishery graduate student wanted to know the population mean length of perch in a local lake. She caught 10 perch, measured their length, and found an average length of 3.6 inches, and a standard error of the sample average length of 0.994 inches. What is the 95% confidence interval for the population mean length of the perch in this lake?

CI = (1.35, 5.85).

A fishery graduate student wanted to know the population mean length of perch in a local lake. She caught all the perch she could and claimed that this was all the perch in the lake. Since she claimed that she had caught the population, she chose to use the z-distribution to calculate a confidence interval for the value of the population mean. What do you think of her choice of distribution?

All of these other answers.

38 eggs were randomly chosen from a grocery store and individually weighed. The average weight was 97.8 grams with a standard deviation weight of 4.2 grams. What is the standard error of the sample average / margin of error for a 90% confidence interval for weight?

Grams = 0.6813 / 1.1494

A researcher was interested in the proportion of the population that was left-handed. He collected 61 random samples of 100 people each and found that an average of 16 people were left-handed, with a standard deviation of 4 people. What is the 95% confidence interval for the population mean number of left-handed people in a sample of 100 people?

CI = (14.97, 17.02)

Click two of the following choices as the reasons why the method of hypothesis testing is needed.
To determine if the alternate hypothesis is appropriate for this statistical situation.
To determine if the null hypothesis is appropriate for this statistical situation.
To learn the reasonableness that the null value is the value of the population mean.
To answer if the null value is less than, or greater than, the population mean.

To learn the reasonableness that the null value is the value of the population mean.
To answer if the null value is less than, or greater than, the population mean.

In the statistical method of hypothesis testing, what is the meaning of the word reasonableness?

How reasonable is the null value the value of the population mean.

In the alternative hypothesis of a hypothesis test, what is the meaning of the words 'less than', or 'greater than'?

Could the value of the population mean be less than, or greater than, the value in the null hypothesis.

In the statistical method of hypothesis testing, choose two of the choices below as the assumptions required to perform a hypothesis test.
The sample average is normally distributed.
The individuals were randomly selected.
The data values are normally distributed.
The sample size is greater than 30 data values

The sample average is normally distributed.
The individuals were randomly selected.

In the statistical method of hypothesis testing, which of the choices below is NOT a descriptive statistic.

Statistic = t₀.

In the statistical method of hypothesis testing, what belief about the population mean does the null hypothesis (H₀) express?

The currently believed value of the population mean.

In the statistical method of hypothesis testing, what belief about the population mean does the alternative hypothesis (H₁) express?
What information does the alternative hypothesis express?

The researcher's believed value of the population mean.

In the statistical method of hypothesis testing, for the value that separates the schematic curve into the acceptance region and the rejection region(s), what is its name / where does it come from?

The critical value / From the z,t-Table.

In the statistical method of hypothesis testing, where does the value of the test statistic come from?

From the t-equation (t₀ = (x̄ − μ) / (s / √n)).

In the statistical method of hypothesis testing, p-value gives the size of which area(s) in a schematic curve?

The tail area(s) from the value of the test statistic

In the statistical method of hypothesis testing, what happens to the Type-II Error when the Type-I Error is decreased?

The Type-II Error increases.

In the science of statistics, why is the method of hypothesis testing called an inferential method of statistics?

It infers whether the researcher's belief is correct from sample information.

In the statistical method of hypothesis testing, how is an inference about a population mean actually made?

By determining if the test statistic is close or far from the null value.

In the statistical method of hypothesis testing, what value determines if the test statistic is close or far / where does it come from?

The critical value / from the z,t-table.

In the statistical method of hypothesis testing, how is the distance between the test statistic and the null value determined?

By converting into z,t-values and finding the statistical distance.

In the statistical method of hypothesis testing, please select two choices that are the parts of Step A: Abstract.
Calculate the test statistic from the sample information.
Find the critical value from probability.
Write the researcher's question into two hypotheses.
Get descriptive statistics from the sample data values.

Write the researcher's question into two hypotheses.
Get descriptive statistics from the sample data values.

In the statistical method of hypothesis testing, where does the null value NOT come from?

From results from the sample.

In the statistical method of hypothesis testing, please select two choices that are the purposes of Step 1: Theorize.
Use the schematic curve to analyze this situation.
Find the critical value(s) for this situation.
Use the critical value to find the probability for this situation.
Find the appropriate statistical theory for this situation.

Use the schematic curve to analyze this situation.
Find the critical value(s) for this situation.

In the statistical method of hypothesis testing, the critical value comes from the z,t-Table. If the degrees of freedom are known, what two choices below give the needed information to find the correct critical value?
The size of the rejection region (α).
The p-value for this situation.
The value of the test statistic (Z₀, t₀).
The number of tails in this situation.

The size of the rejection region (α).
The number of tails in this situation.

In the statistical method of hypothesis testing, please select two choices that are the parts of Step 2: Analyze.

Find the p-value from the test statistic and the z,t-table.

In the statistical method of hypothesis testing, where does the value of a test statistic come from / where does the p-value come from?

From the z,t-Equation / from the z,t-Table.

In the statistical method of hypothesis testing, please select two choices that are the only possible conclusions of Step 3: Infer.
DO NOT REJECT the null hypothesis.
REJECT the null hypothesis.
REJECT the alternative hypothesis.
DO NOT REJECT the alternative hypothesis.

DO NOT REJECT the null hypothesis.
REJECT the null hypothesis.

In the statistical method of hypothesis testing, please select two choices that are the most commonly used approaches to make an inference about the value of a population mean.
The Significance Level approach.
The Critical Value approach.
The Test Statistic approach.
The p-Value approach.

The Critical Value approach.
The p-Value approach.

In the statistical method of hypothesis testing, what exactly does the conclusion to Reject the null hypothesis mean?

That the alternative hypothesis is closer to the truth.

For the statistical method of hypothesis testing, what is the appropriate situation to use the t-distribution?

When the population standard deviation is not known.

For the statistical method of hypothesis testing, what are the degrees of freedom to use when 48 Tasmanian devils were measured?

DF = 47.

For the statistical method of hypothesis testing, using the information below, what is the magnitude of the critical value for a left-tail hypothesis test? α = 0.10 ; n = 13 ; x̄ = 14.10 ; s = 11.93

CV = 1.356.

For the statistical method of hypothesis testing, using the information below, what is the magnitude of the critical value for a two-tail hypothesis test? α = 0.10 ; n = 13 ; x̄ = 14.10 ; s = 11.93

CV = 1.782.

For the statistical method of hypothesis testing, using the information below, what is the value of the test statistic for a two-tail hypothesis test? α = 0.02 ; μ = 8.0 ; n = 15 ; x̄ = 7.3 ; s = 1.8

t₀ = −1.51

α = 0.05 ; μ = 1.1 ; n = 25 ; x̄ = 1.8 ; s = 0.8 ?For the statistical method of hypothesis testing, using the information below, what is the value of the test statistic for a right-tail hypothesis test?

t₀ = +4.375

For the statistical method of hypothesis testing, select two of the choices below that are the most common approaches to make a conclusion in the Infer step of a hypothesis test.
The p-Value approach.
The Critical Value approach.
The significance value approach.
The test statistic value approach.

The Critical Value approach.
The p-Value approach.

For the statistical method of hypothesis testing, what is the proper conclusion to a hypothesis test using the information below? α = 0.05 ; H₀: μ = 148, H₁: μ ≠ 148 ; CV = ±2.032 ; t₀ = −1.478

Do not reject the null hypothesis as the test statistic is in the acceptance region.