Math 110 Chapter 7, 8, 9
Alternate Hypothesis (H1) - Correct Answer This is the statement you will adopt in the situation in which the evidence
(data) is so strong that you reject H0. A statistical test is designed to assess the strength of the evidence (data)
against the null hypothesis.
Assessing Normality - Correct Answer We will reject the assumption that a population is approximately normal if a
sample has any of the following features:
1. The sample contains an outlier
2. The sample exhibits a large degree of skewness
3. The sample has more than 1 distinct mode
Assumptions for a Population Proportion Confidence Interval - Correct Answer 1) We have a Simple Random Sample
(SRS)
2) The Population is at least 20 times as large as the sample
3) The items in the population are divided into two categories
4) The sample must contain at least 10 individuals in each categories
Assumptions for Hypothesis Tests about Proportions - Correct Answer 1. SRS
2. The population is at least 10x the size of the sample (n≤0.05N)
3. The item in the population is divided into 2 categories
4. The values np0 and n(1-p0) are both at least 10 (n*p & n*(1-p) ≥ 10)
Central Limit Theorem Rule of Thumb - Correct Answer The distribution may be approximated with a normal curve
whenever n*p and n*(1-p) are both at least 10.
Checking a Confidence Interval with Population Standard Deviation known - Correct Answer Stat -> Tests -> 7:
ZInterval -> Stats ->Input Data
Chi-Square (χ2) Distribution Characteristics - Correct Answer 1. The χ2 Distributions are not symmetric. They are
skewed to the right
2. Values of the χ2 statistic are always greater or equal to zero. They are never negative.
Conclusions to Hypotheses tests - Correct Answer If:
If H0 is rejected, then we can conclude that H1 is true.
If H0 is NOT rejected, then we can conclude that there is not evidence to conclude that H1 is true.
NOTE: This does not mean that the null hypothesis is true, but that the null hypothesis MIGHT be true.
Confidence Interval - Correct Answer x̄ - m < μ < x̄ + m
Point Estimate - Margin of Error < Mean (Mu) < Point Estimate + Margin of Error
Confidence Interval Assumptions for a Population Mean, σ Known - Correct Answer 1) We have a simple random
sample (SRS)
2) The sample size is large enough (n>30) to apply Central Limit Theorem or the population is approximately normal.
Confidence Intervals when Population Standard Deviation is Unknown - Correct Answer We use a Student's t
distribution
, Since we no longer have the standard deviation, we cannot use Zα/2. We must now use tα/2.
Construct Confidence Intervals for a Population Mean when the Population Standard Deviation is Unknown - Correct
Answer Step 1: Check the assumptions and if it passes, Input all of the data into stats
Step 2: Go to 1-Var Stats and find the Sample mean x̄ and the sample standard deviation s.
Step 3: Find the degrees of freedom (df=n-1) and the critical value tα/2.
Step 4: Compute the standard error s/√n and multiply it by the critical value to get the margin of error
Step 5: Construct the Confidence Interval
Constructing a Confidence Interval for a Chi-Square distribution - Correct Answer Population Variance:
[(n-1)s^2]/χ2(α/2) < σ2 < [(n-1)s^2]/χ2(1-α/2)
Population Standard Deviation =
√([(n-1)s^2]/χ2(α/2)) < σ < √([(n-1)s^2]/χ2(1-α/2))
Continuity Correction - Correct Answer Adjustment made when a discrete random variable is being approximated by
a continuous random variable
Critical Value (Sometimes referred to as Zα/2) - Correct Answer Zα/2 = invNorm(1-((1-confidence level)/2)
step 1) 1 - confidence level = x
step 2) x/2
step 3) invNorm(1-x, 0, 1)
Critical Values of a χ2 distribution - Correct Answer Use the χ2 distribution table to find the values.
The critical values are denoted as χ2(1-α/2) and χ2(α/2)
Example: A machine that fills cereal boxes is supposed to put 20 ounces of cereal in each box. A simple random
sample of 6 boxes is found to contain a sample mean of 20.25 ounces of cereal. It is known from past experience that
the fill weights are normally distributed with a standard deviation of 0.2 ounces. Construct a 90% confidence interval
for the mean fill weight. - Correct Answer First, Check the Assumptions:
1) SRS - Met
2) Large or Normally Distributed - Met
We may construct a confidence interval.
Confidence Level: (1-0.9)/2 = 0.05
Point Estimate: 20.25
Standard Error: 0.2/√6 = 0.08164965809
Critical Value: invNorm(1-0.05, 0, 1) = 1.645
Margin of Error = 1.645 * 0.08164965809 = 0.1343
Confidence Interval:
20.25 - 0.1343 < μ < 20.25 + 0.1343
20.12 < μ < 20.38
Example: A simple random sample of size 10 is drawn from a normal population. Find the critical value tα/2 for a 95%
confidence interval. - Correct Answer Degrees of Freedom = n - 1 = 9
1-0.95 = 0.05
0.05/2 = 0.025
invT(1-0.025, 9) = 2.262
Alternate Hypothesis (H1) - Correct Answer This is the statement you will adopt in the situation in which the evidence
(data) is so strong that you reject H0. A statistical test is designed to assess the strength of the evidence (data)
against the null hypothesis.
Assessing Normality - Correct Answer We will reject the assumption that a population is approximately normal if a
sample has any of the following features:
1. The sample contains an outlier
2. The sample exhibits a large degree of skewness
3. The sample has more than 1 distinct mode
Assumptions for a Population Proportion Confidence Interval - Correct Answer 1) We have a Simple Random Sample
(SRS)
2) The Population is at least 20 times as large as the sample
3) The items in the population are divided into two categories
4) The sample must contain at least 10 individuals in each categories
Assumptions for Hypothesis Tests about Proportions - Correct Answer 1. SRS
2. The population is at least 10x the size of the sample (n≤0.05N)
3. The item in the population is divided into 2 categories
4. The values np0 and n(1-p0) are both at least 10 (n*p & n*(1-p) ≥ 10)
Central Limit Theorem Rule of Thumb - Correct Answer The distribution may be approximated with a normal curve
whenever n*p and n*(1-p) are both at least 10.
Checking a Confidence Interval with Population Standard Deviation known - Correct Answer Stat -> Tests -> 7:
ZInterval -> Stats ->Input Data
Chi-Square (χ2) Distribution Characteristics - Correct Answer 1. The χ2 Distributions are not symmetric. They are
skewed to the right
2. Values of the χ2 statistic are always greater or equal to zero. They are never negative.
Conclusions to Hypotheses tests - Correct Answer If:
If H0 is rejected, then we can conclude that H1 is true.
If H0 is NOT rejected, then we can conclude that there is not evidence to conclude that H1 is true.
NOTE: This does not mean that the null hypothesis is true, but that the null hypothesis MIGHT be true.
Confidence Interval - Correct Answer x̄ - m < μ < x̄ + m
Point Estimate - Margin of Error < Mean (Mu) < Point Estimate + Margin of Error
Confidence Interval Assumptions for a Population Mean, σ Known - Correct Answer 1) We have a simple random
sample (SRS)
2) The sample size is large enough (n>30) to apply Central Limit Theorem or the population is approximately normal.
Confidence Intervals when Population Standard Deviation is Unknown - Correct Answer We use a Student's t
distribution
, Since we no longer have the standard deviation, we cannot use Zα/2. We must now use tα/2.
Construct Confidence Intervals for a Population Mean when the Population Standard Deviation is Unknown - Correct
Answer Step 1: Check the assumptions and if it passes, Input all of the data into stats
Step 2: Go to 1-Var Stats and find the Sample mean x̄ and the sample standard deviation s.
Step 3: Find the degrees of freedom (df=n-1) and the critical value tα/2.
Step 4: Compute the standard error s/√n and multiply it by the critical value to get the margin of error
Step 5: Construct the Confidence Interval
Constructing a Confidence Interval for a Chi-Square distribution - Correct Answer Population Variance:
[(n-1)s^2]/χ2(α/2) < σ2 < [(n-1)s^2]/χ2(1-α/2)
Population Standard Deviation =
√([(n-1)s^2]/χ2(α/2)) < σ < √([(n-1)s^2]/χ2(1-α/2))
Continuity Correction - Correct Answer Adjustment made when a discrete random variable is being approximated by
a continuous random variable
Critical Value (Sometimes referred to as Zα/2) - Correct Answer Zα/2 = invNorm(1-((1-confidence level)/2)
step 1) 1 - confidence level = x
step 2) x/2
step 3) invNorm(1-x, 0, 1)
Critical Values of a χ2 distribution - Correct Answer Use the χ2 distribution table to find the values.
The critical values are denoted as χ2(1-α/2) and χ2(α/2)
Example: A machine that fills cereal boxes is supposed to put 20 ounces of cereal in each box. A simple random
sample of 6 boxes is found to contain a sample mean of 20.25 ounces of cereal. It is known from past experience that
the fill weights are normally distributed with a standard deviation of 0.2 ounces. Construct a 90% confidence interval
for the mean fill weight. - Correct Answer First, Check the Assumptions:
1) SRS - Met
2) Large or Normally Distributed - Met
We may construct a confidence interval.
Confidence Level: (1-0.9)/2 = 0.05
Point Estimate: 20.25
Standard Error: 0.2/√6 = 0.08164965809
Critical Value: invNorm(1-0.05, 0, 1) = 1.645
Margin of Error = 1.645 * 0.08164965809 = 0.1343
Confidence Interval:
20.25 - 0.1343 < μ < 20.25 + 0.1343
20.12 < μ < 20.38
Example: A simple random sample of size 10 is drawn from a normal population. Find the critical value tα/2 for a 95%
confidence interval. - Correct Answer Degrees of Freedom = n - 1 = 9
1-0.95 = 0.05
0.05/2 = 0.025
invT(1-0.025, 9) = 2.262
