Chapter 9: One Sample T-Test

P- value

Probability of obtaining a test statistic equal to or more extreme than the observed sample value given H0 is true - The p-value is also called the observed level of significance - It is the smallest value of a for which H0 can be rejected

critical values correspond to

level of significance

Holding the confidence level constant, if we increase the sample size, this will result in a

narrower interval. - If we decrease the sample size, this will result in a wider interval

Holding the sample size constant, if we decrease our confidence, this will result in a

narrower interval. - If we increase our confidence, this will result in wider interval.

If type 1 probability increases, type two error

decreases

If t statistic is not within region of rejection

do NOT reject the null

Type 1 and 2 errors can NOT happen at the same time but...

they are inversely related.

If t statistic is within region of rejection (tail ends)

reject the null

Null hypothesis: What is the status quo? What is the current situation?

Always has a =, ≤, or ≥

Alternative hypothesis: What are we trying to prove?

Always has ≠, <, or >

To evaluate the normality assumption

- Determine how closely sample statistics match the normal distribution's theoretical properties. - Construct a histogram or boxplot

Compare the p-value with α

-If p-value < α , reject H0 -If p-value α , do not reject H0 -If the p-value is low then H0 must go

In order to reduce the number of Type1 errors, where do you set your significance level?

.05

Industry standard says level of significance, α, is

.05

For a two-tail test for the mean, σ known:

1) Convert sample statistic (X) to test statistic (Zstat) 2) Determine the critical Z values for a specified level of significance (a) from a table or by using computer software ** Decision Rule: If the test statistic falls in the rejection region, reject H0 ; otherwise do not reject H0

An appliance manufacturer claims to have developed a compact microwave that consumes a mean of more than 250 W. From previous studies, it is believed that power consumption for microwave ovens is normally distributed with a population standard deviation of 15 W. A consumer group has decided to try to discover if the claim appears to be true. They take a sample of 20 microwave ovens and find that they consume a mean of 257.3 W. 1) The population of interest is: 2) The parameter of interest is: 3) State the null hypothesis: 4) State the alternative hypothesis: 5) With a significance level of .05, the critical value would be Z = 1.645. The value of the test statistic is calculated to be 2.18. The p-value of the test is .0148. - What can the consumer group conclude?

1) The population of interest is: microwave ovens 2) The parameter of interest is: The mean power consumption of all such microwave ovens 3) State the null hypothesis: H0: µ ≤ 250 4) State the alternative hypothesis: H1: µ > 250 5) All of the following are valid conclusions - There is sufficient evidence that the mean power consumption is not less than or equal to 250 - There is sufficient evidence that the mean power consumption is greater than 250 - There is enough evidence that the manufacturer's claim is false

The quality control engineer for a furniture manufacturer is interested in the mean amount of force necessary to produce cracks in the stressed oak furniture. She performs a two-tail test of the null hypothesis that the mean for the stressed oak furniture is 650. The calculated Z test statistic is a positive number that leads to a p-value of .080 for the test. 1) If the test was performed with a level of significance of .10, the null hypothesis would (be/not be) rejected. 2) What can she conclude? 3) If the test was performed with a level of significance of .05, the null hypothesis would (be/not be) rejected. 4) What can she conclude?

1) would BE 2) If the test is performed with a level of significance of .10, there is sufficient evidence to conclude that the mean amount of force necessary cracks is not 650. - There is enough evidence that the null is not true. 3)would NOT be 4) If the test is performed with a level of significance of .05, there is insufficient evidence to show that the mean amount of force necessary to produce cracks is not 650. -There is not enough evidence to show that the null is not true.

Willy Wonka is developing a new Everlasting Gobstopper. The current time it takes for a gobstopper to dissolve has a normal distribution with a mean of 7.3 minutes and a standard deviation of 1.4 minutes. This new gobstopper is such that it should be normally distributed with the same standard deviation, but the time it takes to dissolve may be different than the current gobstopper. A sample size of 46 new gobstoppers results in a sample mean of 8.4 minutes. 1. Write the null and alternative hypotheses: 2. If we conclude that the gobstopper takes either longer or shorter to dissolve than the current one, when in reality, it does not, what type of error is this? 3. If we conclude that the gobstopper doesn't have a different dissolve time than the current one, when in reality, it does, what type of error is this? 4. Suppose that the true time it takes for a new gobstopper to dissolve is 8.32 minutes. How large of a sample size should we take to assume that the sampling distribution is normal? 5. For this new gobstopper, we take a sample of 35 new gobstoppers and find that the average time to dissolve is 8.2 minutes. A 95% confidence interval is calculated: [7.9, 8.5] Interpret this interval.

1. (Null) H0: μ = 7.3 (Alternative) H1: μ ≠ 7.3 2. Type 2 error bc you are NOT rejecting the null, when you really should have. 3. Type 1 error bc you are rejecting the null when you shouldn't have. 4. 30+, according to the central limit theorem 5. We are 95% confident that the true mean dissolve time for the new recipe is between 7.9 and 8.5. - If we were to take all possible samples of 35 gobstoppers and create an interval for each, then 95% of the intervals would contain the true population parameter.

Test the claim that the true mean diameter of a manufactured bolt is 30mm. (Assume σ = 0.8) 1. State the appropriate null and alternative hypotheses. 2. Specify the desired level of significance and the sample size 3. Determine the appropriate technique. 4. Determine the critical values. 5. Collect the data and compute the test statistic

1. Null: H0: μ = 30 Alt. Hypothesis: H1: μ ≠ 30 (This is a two-tail test) 2. Suppose that α = 0.05 and n = 100 are chosen for this test 3. σ is assumed known so this is a Z test. 4. For α = 0.05 the critical Z values are ±1.96 5. Suppose the sample results are n = 100, X = 29.84 (σ = 0.8 is assumed known) THE STATISTIC IS: -2 = Reject the Null (there is sufficient evidence that the mean diameter of a manufactured bolt is not equal to 30)

Chick Fil A wants to update their nutrition facts and needs to know how many calories are 1 box of nuggets (12 count). We find that mean number of calories in a 12 count box is 389.7 with a standard deviation of 6.41. 1. Identify the population of interest, the parameter of interest, the sample, and the statistic: 2. Suppose that that the mean of the sampling distribution (mean of all means) is equal to 390. What is the population parameter equal to? 3. Complete the sentence: We can assume that the sampling distribution of the mean is normal in this particular scenario because... 4. If we assume that calories of chicken nugget boxes are normally distributed, then our sampling distribution of the mean will be normal if... 5. Identify the margin of error: 6. Which of the following conclusions can we make? Mark all that apply. a. We are 95% confident that the true number of calories in a 12 count nugget box is somewhere between 387.4 and 392 calories. b. 95% of the boxes will have between 387.4 and 392 calories. c. 95% of the samples size of 30 will have a mean number of calories between 387.4 and 392. d. It is possible that the true number of calories of a nugget box is NOT between 387.4 and 392. e. If we were to take all possible sample size of 30 boxes, find the mean, and calculate a confidence interval for each, then 95% of the intervals would contain the true number of calories. f. We are 95% confident that all samples will give us a mean between 387.4 and 392.

1. Population of interest: Nugget boxes Parameter of interest: # of calories Sample: 30 boxes Statistic: 389.7 calories 2. 390 3. We have a sample size greater than or equal to 30 4. Regardless of the sample size 5. +/- 2.3 6. d. It is possible that the true number of calories of a nugget box is NOT between 387.4 and 392. e. If we were to take all possible sample size of 30 boxes, find the mean, and calculate a confidence interval for each, then 95% of the intervals would contain the true number of calories. - Hint: The confidence level (95%) should always refer to either INTERVALS or CONFIDENCE. Notice that b. the 95% refers to boxes, c. 95% refers to sample sizes. This is incorrect! Option f is incorrect because confidence intervals are used to make conclusions about parameters. There isn't a need to create confidence interval for a sample. We have that information!

Microsoft Excel was used on a set of data involving the number of defective items found in a random sample of 46 cases of light bulbs produced during a morning shift at a plant. A manager wants to know if the mean number of defective bulbs per case is greater than 20 during the morning shift. She will make her decision using a test with a level of significance of 0.10. The following information was extracted from the Microsoft Excel output for the sample of 46 cases: n = 46 Arithmetic Mean = 28.00 Standard Deviation = 25.92 Standard Error = 3.82; Null Hypothesis: H0 : μ ≤ 20 α = 0.10 df = 45 T Test Statistic = 2.09; One-Tail Test Upper Critical Value = 1.3006 p-value = 0.021 1. What should the manager decide? 2. T/F: The manager can conclude that there IS sufficient evidence to show that the mean number of defective light bulbs is greater than 20 during the morning shift.

1. Reject the null 2. TRUE

6 Steps in Hypothesis Testing

1. State the null hypothesis, H0 and the alternative hypothesis, H1 2. Choose the level of significance, (α ), and the sample size, (n). - The level of significance is based on the relative importance of Type I and Type II errors 3. Determine the appropriate test statistic and sampling distribution 4 .Determine the critical values that divide the rejection and non-rejection regions 5. Collect data and compute the value of the test statistic 6. Make the statistical decision and state the conclusion. - If the test statistic falls into the non-rejection region, do not reject the null hypothesis H0. - If the test statistic falls into the rejection region, reject the null hypothesis. ** Express the conclusion in the context of the problem

The 5 Step p-value approach to Hypothesis Testing

1. State the null hypothesis, H0 and the alternative hypothesis, H1 2. Choose the level of significance, a, and the sample size, n. The level of significance is based on the relative importance of the risks of a type I and a type II error. 3. Determine the appropriate test statistic and sampling distribution 4. Collect data and compute the value of the test statistic and the p-value P(Z><#)= p-value p-value + p-value = answer 5. Make the statistical decision and state the managerial conclusion. If the p-value is < α then reject H0, otherwise do not reject H0. State the managerial conclusion in the context of the problem

We want to test with an alpha (α) of 5%. For a sample size of 40, our critical value is +/- 2.021. Our test statistic is calculated to be 2.23 with a p-value of .0316. 1. Should we reject or not reject the null: What should we conclude? a. Don't change the nutrition information listed on the menu. We have sufficient evidence to conclude that the number of calories in a nugget box is not 390. b. Don't change the nutrition information listed on the menu. We have insufficient evidence to conclude that the number of calories in a nugget box is not 390. c. Change the nutrition information listed on the menu. We have sufficient evidence to conclude that the number of calories in a nugget box is not 390. d. Change the nutrition information listed on the menu. We have insufficient evidence to conclude that the number of calories in a nugget box is not 390.

1. The t statistic is in the region of rejection The p value is less than the alpha 2. c. Change the nutrition information listed on the menu. We have sufficient evidence to conclude that the number of calories in a nugget box is not 390.

The marketing manager for an automobile manufacturer is interested in determining the proportion of new compact-car owners who would have purchased a GPS navigation system if it had been available for an additional cost of $300. The manager believes from previous information that the proportion is 0.30. Suppose that a survey of 200 new compact-car owners is selected and 79 indicate that they would have purchased the GPS navigation system. 1. If you were to conduct a test to determine whether there is evidence that the proportion is different from 0.30 and decided NOT to reject the null hypothesis, what conclusion could you reach? 2. If you were to conduct a test to determine whether there is evidence that the proportion is different from 0.30 at a 1% level of significance

1. There is not sufficient evidence that the proportion is not 0.30. 2. there is sufficient evidence that the proportion is different from 0.30.

2 ways to make a decision on whether or not a null hypothesis should be rejected.

1. compare test statistic to critical values. 2. compare P-values to level of significance.

Identify each of the scenarios as either a Type 1, Type 2, or correct decision. 1. We conclude that the number of calories in a nugget box is not 390, when in reality, there are not 390 calories in a box. 2. We conclude that the number of calories in a nugget box is not 390, when in reality, there are 390 calories in a box. 3. We conclude that the number of calories in a nugget box is 390, when in reality, there are 390 calories in a box. 4. We conclude that the number of calories in a nugget box is 390, when in reality, there are not 390 calories in a box.

1. correct 2. type 1 error 3. correct 4. type 2 error

If the length of broomstick is normally distributed with a mean of 42 inches and a standard deviation of 2.5 inches, what % of broomsticks are less than 39.9 inches (The cumulative standard normal distribution table indicates a z value of -.84 for 20%)

20%

If the probability of going home on Easter is .5, and the probability of going on an egg hunt is .5, what is the probability of going home and going on an egg hunt?

25% .5x.5=.25 (25%)

According to the empirical rule, if data form a normal distribution _____ % of the observations will be contained within 2 standard deviations around the mean

95

Which of the following would be an appropriate null hypothesis? A. The mean of a population is equal to 55. B. The mean of a sample is equal to 55. C. The mean of a population is greater than 55. D. Only (a) and (c) are true.

A. The mean of a population is equal to 55.

How many tissues should the Kimberly Clark Corporation package of Kleenex contain? Researchers determined that 60 tissues is the mean number of tissues used during a cold. Suppose a random sample of 100 Kleenex users yielded the following data on the number of tissues used during a cold: X= 52, S = 22. Suppose the test statistic DOES fall in the rejection region at α = 0.05. Which of the following decision is correct?

At α = 0.10, there is sufficient evidence to conclude that the average number of tissues used during a cold is NOT 60 tissues.

Which of the following types of samples would be least effective if you wanted to make valid inferences from a sample to a population? A. simple random sample B. judgment sample C. stratified sample D. clustered sample

B. judgment sample Bc you would want to use a probability sample instead of a non-probability sample (which a judgment sample is)

Which of the following statements is NOT true about the level of significance in a hypothesis test? A)The higher the significance level, the more likely you are to reject the null hypothesis. B)The significance level is another name for the probability of a Type I error C) The significance level is another name for the probability of a Type II error D) The notation used for is the significance level is α

C) The significance level is another name for the probability of a Type II error

Which of the following would be an appropriate alternative hypothesis? A. The mean of a population is equal to 55. B. The mean of a sample is equal to 55. C. The mean of a population is greater than 55. D. The mean of a sample is greater than 55.

C. The mean of a population is greater than 55.

Another name for the mean of a probability distribution is its A. variance B. standard deviation C. standard error D. expected value

D. expected value

The owner of the Chick Fil A on Georgia claims that his nugget boxes do not have 390 calories. He takes a sample of 40 boxes and finds that the mean calories is 396 calories. Write the null and alternative hypotheses:

H0: µ = 390 H1: µ ≠ 390

A drug company in considering marketing a new local anesthetic. The effective time of the anesthetic the drug company in currently producing has a normal distribution with a moan of 7.4 minutes with a standard deviation of 12 minutes. The chemistry of the new anesthetic is such that the effective time should be normally distributed with the same standard deviation, but the mean effective time may be lower. If it Is lower, the drug company will market the new anesthetic; otherwise, they will continue to produce the older one. A sample of size 36 results in a sample mean of 7.1. A hypothesis test will be done to help make the decision. The appropriate hypotheses are

H0: µ = 7.4 H1: µ < 7.4

A pizza chain is considering opening a new store in an area that currently does not have any such stores. The chain will open if there is evidence that more than 5,000 of the 20,000 households in the area have a favorable view of its chain. It conducts a telephone poll of 300 randomly selected households in the area and finds that 96 have a favorable view. State the test of hypothesis that is of interest to the pizza chain.

H0: π ≤ 0.25 versus H1: π > 0.25

The p value is the probability of your t statistic. It is the probability you got this result given the null.

If p < α, reject the null. If p > a, do not reject the null. If you reject the null, you have sufficient evidence to say the null is not true. If you do not reject the null, you have insufficient evidence to say the null is not true.

If a test of hypothesis has a Type I error probability (alpha) of 0.01. it means that

If the null hypothesis is true, you reject it 1% of the time.

Test it.....

If the sample mean is close to the stated population mean, the null hypothesis is not rejected. If the sample mean is far from the stated population mean, the null hypothesis is rejected. How far is "far enough" to reject H0? The critical value of a test statistic creates a "line in the sand" for decision making -- it answers the question of how far is far enough.

Alternative Hypothesis H1

Opposite of null hypothesis. - Challenges the status quo - NEVER contains =, or ≤, or ≥ sign - Is generally the hypothesis that the researcher is trying to prove. Example: The mean diameter of a manufactured bold is not = to 30mm (H1: μ ≠ 30)

Do You Ever Truly Know σ?

Probably not! In virtually all real world business situations, σ is not known. If there is a situation where σ is known then µ is also known (since to calculate σ you need to know µ.) If you truly know µ there would be no need to gather a sample to estimate it.

Suppose you run a one sample hypothesis test for for following H0: μ = 16 at α =.05. We find a p value of .035. What should we conclude?

Reject the null. We have sufficient evidence that the mean is NOT equal to 16.

Null Hypotheses (H0)

States the claim or assertion to be tested is always about a population parameter, NOT about a sample statistic Claims that there is NO difference in the data gathered to answer the research question Begins with the assumption that the null hypothesis is TRUE. - Similar to the notion of innocent until proven guilty. - Refers to the status quo or historical value - Always contains =, or ≤, or ≥ sign - May or may not be rejected Example: The mean diameter of a manufactured bolt is 30mm (Ho: μ=30)

T/F A narrow confidence interval is more useful than a wide one

TRUE

α is your

Type 1 error rate, chosen by you as a statistician!

Risks in decision making using hypothesis testing:

Type I Error: - Reject a TRUE null hypothesis - A type I error is a "false alarm" - The probability of a Type I Error is α (alpha) ** Called level of significance of the test ** Set by researcher in advance Type II Error: - FAILURE to reject a FALSE null hypothesis - Type II error represents a "missed opportunity" - The probability of a Type II Error is β **Type I and Type II errors cannot happen at the same time - A Type I error can only occur if H0 is true - A Type II error can only occur if H0 is false

two-tailed test

a hypothesis test in which the research hypothesis does not indicate a direction of the mean difference or change in the dependent variable, but merely indicates that there will be a mean difference you are looking at both ends of the distribution

Hypothesis

claim (assertion) about a population parameter Example: - population mean: The mean monthly cell phone bill in this city is μ = $42. - population portion: The proportion of adults in this city with cell phones is π = 0.68

To use the t-test, one must assume the population is

normal •As long as the sample size is not very small and the population is not very skewed, the t-test can be used.

If you are interested in a particular direction, you can use

one-tailed test

Which type of graph is best for presenting 2 numerical variables?

scatter plot

If you reject the null, you are at risk for what type of error?

type 1

Your critical value tells you

when "that's too far" from your null.

What unique symbol is used in a population mean?

μ

What unique symbol is used in a population proportion?

π

Hypothesis Testing When σ (population standard deviation) is Unknown

• If the population standard deviation (σ) is unknown, you instead use the sample standard deviation S. • Because of this change, you use the t distribution instead of the Z distribution to test the null hypothesis about the mean. • When using the t distribution you must assume the population you are sampling from follows a normal distribution. • All other steps, concepts, and conclusions are the same.

one-tailed test

• In many cases, the alternative hypothesis focuses on a particular direction • There is only ONE critical value, since the rejection area is in only ONE tail A hypothesis test in which rejection of the null hypothesis occurs for values of the test statistic in one tail of its sampling distribution. IF H0: μ ≥ 3 OR H1: μ < 3 This is a LOWER-tail test since the alternative hypothesis is focused on the lower tail below the mean of 3 IF H0: μ ≤ 3 OR H1: μ > 3 This is an UPPER-tail test since the alternative hypothesis is focused on the upper tail above the mean of 3

Null Hypothesis: Statistical Significance vs Practical Significance

• Statistically significant results (rejecting the null hypothesis) are not always of practical significance -This is more likely to happen when the sample size gets very large • Practically important results might be found to be statistically insignificant (failing to reject the null hypothesis) -This is more likely to happen when the sample size is relatively small

•Claim: The population mean age is 50. -H0: μ = 50, H1: μ ≠ 50 •Sample the population and find the sample mean.

•Suppose the sample mean age was X = 20. •This is significantly lower than the claimed mean population age of 50. •If the null hypothesis were true, the probability of getting such a different sample mean would be very small, so you reject the null hypothesis . •In other words, getting a sample mean of 20 is so unlikely if the population mean was 50, you conclude that the population mean must not be 50.

Possible errors in hypothesis test decision making

•The confidence coefficient (1-α) is the probability of not rejecting H0 when it is true. •The confidence level of a hypothesis test is (1-α)*100%. •The power of a statistical test (1-β) is the probability of rejecting H0 when it is false.