Bana Final Chapter 9 and 10
Refer to Question 2, what is the p-value associated with the hypothesis test? Round your answer to three decimal places.
0.006
Refer to Questions 2 & 3. The p-value is
0.0228
Refer to Questions 1 & 2 and round your answer to three decimals. The p-value is
0.046
Refer to Question 5 and round your answer to three decimals. The p-value is
0.07
CHAOTER 10
CHAPTER 10
If the probability of a Type I error (α) is .05, then the probability of a Type II error (β) must be
Cannot be computed
Hypothesis Tests About m1 - m2: s1 and s2 Known
Look at slides
Rejection Rule: p-vaue Approach
Reject H0 if p -value < a
Hypothesis testing
can be used to determine whether a statement about the value of a population parameter should or should not be rejected
null hypothesis
denoted by H0 , is a tentative assumption about a population parameter.
alternative hypotheis
denoted by Ha, is the opposite of what is stated in the null hypothesis.
sampling distribution of x1 - x2
where: 1 = standard deviation of population 1 2 = standard deviation of population 2 n1 = sample size from population 1 n2 = sample size from population 2
If the null hypothesis is not rejected at the 5% level of significance, it
will also not be rejected at 1% level
If the level of significance of a hypothesis test is raised from .01 to .05, the probability of a Type II error
will decrease
A random sample of 100 people was taken. Eighty-five of the people in the sample favored Candidate A. We are interested in determining whether or not the proportion of the population in favor of Candidate A is significantly more than 80%. The p-value is
.1056
A random sample of 100 people was taken. Eighty-five of the people in the sample favored Candidate A. We are interested in determining whether or not the proportion of the population in favor of Candidate A is significantly more than 80%. The test statistic is
1.25
Refer to Question 1 and round your answer to three decimals. The value of the test statistic is
2.00
Refer to Question 2. The value of the test statistic is
2.00
A random sample of 100 people was taken. Eighty-five of the people in the sample favored Candidate A. We are interested in determining whether or not the proportion of the population in favor of Candidate A is significantly more than 80%. At the .05 level of significance, it can be concluded that the proportion of the population in favor of candidate A is
80%
Type 1 Error
A Type I error is rejecting H0 when it is true. The probability of making a Type I error when the null hypothesis is true as an equality is called the level of significance. Applications of hypothesis testing that only control for the Type I error are often called significance tests.
p value approach to two tailed hypothesis testing
Compute the p-value using the following three steps 1. Compute the value of the test statistic z 2. If z is in the upper tail (z > 0), compute the probability that z is greater than or equal to the value of the test statistic. If z is in the lower tail (z < 0), compute the probability that z is less than or equal to the value of the test statistic. 3. Double the tail area obtained in step 2 to obtain the p-value. The rejection rule: Reject H0 if the p-value <
A school's newspaper reported that the proportion of students majoring in business is at least 30%. You plan on taking a sample to test the newspaper's claim. The correct set of hypotheses is
H0: p .30 Ha: p < .30.
Inference About Means and Proportions with Two Populations
Inferences About the Difference Between Two Population Means: s 1 and s 2 Known Inferences About the Difference Between Two Population Means: s 1 and s 2 Unknown Inferences About the Difference Between Two Population Means: Matched Samples
Two Population Means: s 1 and s 2 Known
Interval Estimation of m 1 - m 2 Hypothesis Tests About m 1 - m 2
Type 2 Error
It is difficult to control for the probability of making a Type II error Statisticians will state "do not reject H0" and rather than "accept H0" to acknowledge the possibility of Type II error.
Guidelines for interepting p-values
Less than .01 Overwhelming evidence to conclude H0 is false. Between .01 and .05 Strong evidence to conclude H0 is false. Between .05 and .10 Weak evidence to conclude H0 is false. Greater than .10 Insufficient evidence to conclude H0 is false
Estimating the difference between two population means
Let 1 equal the mean of population 1 and 2 equal the mean of population 2. The difference between the two population means is 1 - 2. To estimate 1 - 2, we will select a simple random sample of size n1 from population 1 and a simple random sample of size n2 from population 2. Let 𝑥 ̅_1 equal the mean of sample 1 and 𝑥 ̅_2 equal the mean of sample 2. The point estimator of the difference between the means of the populations 1 and 2 is 𝑥 ̅_1−𝑥 ̅_2.
Point Estimate of U1 - U2
Point estimate of 1 - 2 = 𝑥 ̅_1−𝑥 ̅_2 where: 1 = mean distance for the population of Par, Inc. golf balls 2 = mean distance for the population of Rap, Ltd. golf balls
Confidence Interval Approach to Two-Tailed Tests About a Population Mean
Select a simple random sample from the population and use the value of the sample mean 𝑥 ̅ to develop the confidence interval for the population mean . (Confidence intervals are covered in Chapter 8.) If the confidence interval contains the hypothesized value 0, do not reject H0. Otherwise, reject H0. (Actually, H0 should be rejected if 0 happens to be equal to one of the end points of the confidence interval.)
Steps of Hypothesis Testing
Step 1. Develop the null and alternative hypotheses Step 2. Specify the level of significance . Step 3. Collect the sample data and compute the value of the test statistic p-Value Approach Step 4. Use the value of the test statistic to compute the p-value. Step 5. Reject H0 if p-value < a. Critical Value Approach Step 4. Use the level of significance a to determine the critical value and the rejection rule Step 5. Use the value of the test statistic and the rejection rule to determine whether to reject H0.
Critical Value Approach to Two-Tailed Hypothesis Testing
The critical values will occur in both the lower and upper tails of the standard normal curve. Use the standard normal probability distribution table to find z/2 (the z-value with an area of a/2 in the upper tail of the distribution). The rejection rule is: Reject H0 if z < -z/2 or z > z/2.
p -Values and the t Distribution
The format of the t distribution table provided in most statistics textbooks does not have sufficient detail to determine the exact p-value for a hypothesis test. However, we can still use the t distribution table to identify a range for the p-value. An advantage of computer software packages is that the computer output will provide the p-value for the t distribution.
Power of the test
The probability of correctly rejecting H0 when it is false is called the power of the test. For any particular value of m, the power is 1 - b. We can show graphically the power associated with each value of m; such a graph is called a power curve. (See next slide.)
Critical Value Approach to One-Tailed Hypothesis Testing
The test statistic z has a standard normal probability distribution. We can use the standard normal probability distribution table or function to find the z-value with an area of a in the lower (or upper) tail of the distribution The value of the test statistic that established the boundary of the rejection region is called the critical value for the test. The rejection rule is: Lower tail: Reject H0 if z < -z Upper tail: Reject H0 if z > z
If a hypothesis test leads to the rejection of the null hypothesis,
Type 1 error may have been committed
What type of error occurs if you fail to reject H0 when, in fact, it is not true?
Type 2
The power curve provides the probability of
correctly rejecting the null hypothesis
Refer to Question 6. If alpha is .05, the correct conclusion is
do not reject the null hypothesis
For a given level of Type I error, if we want to decrease the level of Type II error, the sample size must
increase
The sum of the values of α and β
is not needed in hypothesis testing
p value
is the probability, computed using the test statistic, that measures the support or lack of support provided by the sample for the null hypothesis. The p-value is the probability of being wrong if the null hypothesis is rejected. If the p-value is less than or equal to the level of significance , the value of the test statistic is in the rejection region. Reject H0 if the p-value <
Rejection Rule: Critical Value Approach
look at slide
Generally, the ________ sample procedure for inferences about two population means provides better precision than the _______ sample approach
matched independent
When each data value in one sample is matched with a corresponding data value in another sample, the samples are known as
matched samples
p value
must be a number between zero and 1
When the null hypothesis is not rejected, it is
possible a type 2 error has occurred
Refer to Question 3. If alpha is .05, the correct conclusion is
reject
Refer to Question 3. If alpha is .05, the correct conclusion is
reject the null hypothesis
Refer to Question 4. If alpha is .05, the correct conclusion is
reject the null hypothesis
More evidence against H0 is indicated by
smaller p values
A carpet company advertises that it will deliver your carpet within 15 days of purchase. A sample of 49 past customers is taken. The average delivery time in the sample was 16.2 days and the standard deviation was 5.6 days.
the average delivery time is less than or equal to 15 days
The average price of homes sold in the U.S. in the past year was $220,000. A random sample of 81 homes sold this year showed an average price of $210,000. It is known that the standard deviation of the population is $36,000. Has the price decreased?
the average house price is greater than or equal to $220,000
A lathe is set to cut bars of steel into lengths of 6 centimeters. The lathe is considered to be in perfect adjustment if the average length of the bars it cuts is 6 centimeters. A sample of 121 bars is selected randomly and measured. It is determined that the average length of the bars in the sample is 6.08 centimeters. The population standard deviation is 0.44 centimeters.
the average length equals 6 centimeter
For a given sample size in hypothesis testing,
the smaller the type 1 error the larger the type 2 error will be
The sales of a grocery store had an average of $8,000 per day. The store introduced several advertising campaigns in order to increase sales. To determine whether or not the advertising campaigns have been effective in increasing sales, a sample of 64 days of sales was selected. It was found that the average was $8,300 per day. From past information, it is known that the standard deviation of the population is $1,200. the correct null hypothesis is
u is less than or equal to 8000
hypothesis testing procedure
uses data from a sample to test the two competing statemnts indicated by H0 and Ha
Expected Value
𝐸(𝑥 ̅_1−𝑥 ̅_2)=𝜇_1−𝜇_2
two tailed
𝐻_0: 𝜇 = 𝜇_0 𝐻_𝑎: 𝜇 ≠ 𝜇_0
one tailed lower tail
𝐻_0: 𝜇 ≥ 𝜇_0 𝐻_𝑎: 𝜇 < 𝜇_0
one tailed upper tail
𝐻_𝑎: 𝜇 > 𝜇_0 𝐻_0: 𝜇 ≤ 𝜇_0
test staistic
𝑡=(𝑥 ̅−𝜇_0)/(𝑠∕√𝑛)
Interval Estimate
𝑥 ̅_1−𝑥 ̅_2±𝑧_(𝛼/2) √((𝜎_1 )^2/𝑛_1 +(𝜎_2 )^2/𝑛_2 ) where: 1 - is the confidence coefficient.
Tests About a Population Proportion test statistic
𝑧=(𝑝 ̅−𝑝_0)/𝜎_𝑝 ̅ 𝜎_𝑝 ̅ =√((𝑝_0 (1−𝑝_0 ))/𝑛)
Standard Deviation (Standard Error)
𝜎_(𝑥 ̅_1−𝑥 ̅_2 )=√((𝜎_1 )^2/𝑛_1 +(𝜎_2 )^2/𝑛_2 )
