Hypothesis Tests One Sample Means I tell if

Hypothesis Tests One Sample Means

I tell if they really Example. How 2: Acan government agency are underweight? has received numerous complaints that a particular Hypothesis test restaurant has been selling will help me underweight decide! Takehamburgers. a sample & find x. The restaurant advertises that it’s patties. Butare how“adoquarter I know ifpound” this x is(4 one that I expect to happen or is it one ounces). that is unlikely to happen?

What are hypothesis tests? Calculations that tell us if a value occurs by random chance or not – if it is statistically significant Is it. . . – a random occurrence due to variation? – a biased occurrence due to some other reason?

Nature of hypothesis tests How does a murder trial work? • First begin by supposing the “effect” NOTthat present First - is assume the innocent • Next, person see ifisdata provides Then – must have sufficient evidence against the evidence to prove guilty supposition Hmmmmm … Example: murder Hypothesis tests use the same process! trial

Steps: Notice the steps are the same except we add hypothesis statements – which you will learn today 1) Assumptions 2) Hypothesis statements & define parameters 3) Calculations 4) Conclusion, in context

Assumptions for z-test (t-test): YEA – • • These are the same Have an SRS of context assumptions as confidence intervals!! Distribution is (approximately) normal – Given – Large sample size – Graph data • s is known (unknown)

Example 1: Bottles of a popular cola are supposed to bottles contain 300 m. L of • Have an SRS of cola. There is some variation from • Sampling distribution is approximately normalto because is bottle. the. Anboxplot inspector, who symmetrical suspects that the bottler is under • s is unknown filling, measures the contents of six randomly selected bottles. Are the assumptions met? 299. 4 297. 7 298. 9 300. 2 297 301

Writing Hypothesis statements: • Null hypothesis – is the statement being tested; this is a statement of “no effect” or “no difference” H 0: • Alternative hypothesis – is the statement that we suspect is true Ha:

The form: Null hypothesis H 0: parameter = hypothesized value Alternative hypothesis Ha: parameter > hypothesized value Ha: parameter < hypothesized value Ha: parameter = hypothesized value

Example 2: A government agency has received numerous complaints that a particular restaurant has been selling underweight hamburgers. The restaurant advertises that it’s patties are “a quarter pound” (4 ounces). State the hypotheses : H 0 : m = 4 Ha : m < 4 Where m is the true mean weight of hamburger patties

Example 3: A car dealer advertises that is new subcompact models get 47 mpg. You suspect the mileage might be overrated. State the hypotheses : H 0: m = 47 Ha: m < 47 Where m is the true mean mpg

Example 4: Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40 -A fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40. If the mean amperage is lower than 40, customers will complain because the fuses require replacement too often. If the amperage is higher than 40, the manufacturer might be liable for damage to an electrical system due to fuse malfunction. State the hypotheses : H 0: m = 40 Ha: m = 40 Where m is the true mean amperage of the fuses

Facts to remember about hypotheses: • ALWAYS refer to populations (parameters) • The null hypothesis for the “difference” between populations is usually equal to zero H 0: mx-y= 0 • The null hypothesis for the correlation (rho) of two events is usually equal to zero. H 0: r= 0

Activity: For each pair of hypotheses, indicate are not Must use parameter (population) x Must bewhich NOT equal! is & a statistics (sample) legitimate explain why p is the population proportion! Must use same r is parameter number as for H 0! population correlation coefficient – but H 0 MUST be “=“ !

P-values • The probability that the test statistic would have a value as extreme or more than what is actually observed In other words. . . is it far out in the tails of the distribution?

Level of Significance Activity

Level of significance • Is the amount of evidence necessary before we begin to doubt that the null hypothesis is true • Is the probability that we will reject the null hypothesis, assuming that it is true • Denoted by a – Can be any value – Usual values: 0. 1, 0. 05, 0. 01 – Most common is 0. 05

Statistically significant – • The p-value is as small or smaller than the level of significance (a) • If p > a, “fail to reject” reject the null hypothesis at the a level. • If p < a, “reject” reject the null hypothesis at the a level.

Facts about p-values: • ALWAYS make decision about the null hypothesis! • Large p-values show support for the null hypothesis, but never that it is true! • Small p-values show support that the null is not true. • Double the p-value for two-tail (=) tests • Never accept the null hypothesis!

Never “accept” the null hypothesis!

At an a level of. 05, would you reject or fail to reject H 0 for the given p-values? a). 03 b). 15 c). 45 d). 023 Reject Fail to reject Reject

Calculating p-values • For z-test statistic – – Use normalcdf(lb, ub) – [using standard normal curve] • For t-test statistic – – Use tcdf(lb, ub, df)

Draw & shade a curve & calculate the p-value: 1) right-tail test t = 1. 6; n = 20 2) left-tail test z = -2. 4; n = 15 3) two-tail test t = 2. 3; n = 25

Writing Conclusions: 1) A statement of the decision being made (reject or fail to reject H 0) & why (linkage) AND 2) A statement of the results in context. (state in terms of Ha)

“Since the p-value < (>) a, I reject (fail to reject) the H 0. There is (is not) sufficient evidence to suggest that Ha. ” Be sure to write Ha in context (words)!

Example 5: Drinking water is considered unsafe if the mean H 0: m = 15 concentration of lead is 15 ppb (parts Ha : m > 15 or greater. Suppose a per billion) t=2. 1 Where m is the true mean concentration community randomly selects of 25 of lead in drinking water and computes Since samples the p-value < a, I reject Ha 0. t-test There is statistic 2. 1. Assume that lead sufficient evidence to suggest that the P-value =of tcdf(2. 1, 10^99, 24) concentrations are of normally mean concentration lead in drinking =. 0232 water is greater thanthe 15 ppb. distributed. Write hypotheses, calculate the p-value & write the appropriate conclusion for a = 0. 05.

Example 6: A certain type of frozen dinners states that the dinner H 0: m 240 = 240 calories. A random contains Ha: of m > 12 240 of these frozen dinners sample t=1. 9 Where m is the mean caloric was selected fromtrue production to see Since the p-value <frozen a, I reject H 0. There is content of the dinners if the caloric content was greater sufficient evidence to suggest that the than stated on the box. The t-test P-value = tcdf(1. 9, 10^99, 11) true mean caloric content of these frozen statistic was calculated to be 1. 9. =. 0420 dinners is greater than 240 calories. Assume calories vary normally. Write the hypotheses, calculate the p-value & write the appropriate conclusion for a = 0. 05.

Formulas: s known: z= m

Formulas: s unknown: t= m

Example 7: The Fritzi Cheese Company buys milk from several suppliers as the essential raw material for its cheese. Fritzi suspects that some producers are adding water to their milk to increase their profits. Excess water can be detected by determining the freezing point of milk. The freezing temperature of natural milk varies normally, with a mean of -0. 545 degrees and a standard deviation of 0. 008. Added water raises the freezing temperature toward 0 degrees, the freezing point of water (in Celsius). The laboratory manager measures the freezing temperature of five randomly selected lots of milk from one producer with a mean of -0. 538 degrees. Is there sufficient evidence to suggest that this producer is adding water to his milk?

Assumptions: SRS? Normal? • I have an SRS of milk from one producer How do you • The freezing temperature of milk is a normal distribution. know? (given) • s is known Do you What are your H 0: m = -0. 545 know s? hypothesis Ha: m > -0. 545 statements? Is there a key word? where m is the true mean freezing temperature of milk Plug values into formula. p-value = normalcdf(1. 9566, 1 E 99)=. 0252 a =. 05 Use normalcdf to calculate p-value.

Compare your p-value to a & make decision Since p-value < a, I reject the null hypothesis. Conclusion: There is sufficient evidence to suggest that the true mean freezing temperature is greater than -0. 545. This suggests that the producer is adding water to the milk. Write conclusion in context in terms of Ha.

Example 8: The Degree of Reading Power (DRP) is a test of the reading ability of children. Here are DRP scores for a random sample of 44 third-grade students in a suburban district: (data on note page) At the a =. 1, is there sufficient evidence to suggest that this district’s third graders reading ability is different than the national mean of 34?

SRS? • I have an SRS of third-graders • Since the sample size is large, the sampling distribution. Normal? is How do you approximately normally distributed know? OR Do you know s? no outliers, the • Since the histogram is unimodal with What are your sampling distribution is approximately normally distributed hypothesis • s is unknown statements? Is H 0: m = 34 where m is the true mean reading there a key word? Ha: m = 34 ability of the district’s third-graders Plug values into formula. p-value = tcdf(. 6467, 1 E 99, 43)=. 2606(2)=. 5212 Use tcdf to calculate p-value. a =. 1

Compare your p-value to a & make decision Since p-value > a, I fail to reject the null hypothesis. Conclusion: There is not sufficient evidence to suggest that the true mean reading ability of the district’s third-graders is different than the national mean of 34. Write conclusion in context in terms of Ha.

Example 9: The Wall Street Journal (January 27, 1994) reported that based on sales in a chain of Midwestern grocery stores, President’s Choice Chocolate Chip Cookies were selling at a mean rate of $1323 per week. Suppose a random sample of 30 weeks in 1995 in the same stores showed that the cookies were selling at the average rate of $1208 with standard deviation of $275. Does this indicate that the sales of the cookies is different from the earlier figure?

Assume: • Have an SRS of weeks • Distribution of sales is approximately normal due to large sample size • s unknown H 0: m = 1323 Ha: m ≠ 1323 where m is the true mean cookie sales per week Since p-value < a of 0. 05, I reject the null hypothesis. There is sufficient to suggest that the sales of cookies are different from the earlier figure.

Example 9: President’s Choice Chocolate Chip Cookies were selling at a mean rate of $1323 per week. Suppose a random sample of 30 weeks in 1995 in the same stores showed that the cookies were selling at the average rate of $1208 with standard deviation of $275. Compute a 95% confidence interval for the mean weekly sales rate. CI = ($1105. 30, $1310. 70) Based on this interval, is the mean weekly sales rate statistically different from the reported $1323?

What do you notice about the decision from the confidence interval & the hypothesis test? What decision would you make on Example 10 if a =. 01? What confidence level would be correct to use? Does that confidence interval provide the same decision? If Ha: m < 1323, what decision would the hypothesis test give at a =. 05? a =. 01? Now, what confidence levels are appropriate for this alternative hypothesis?

Matched Pairs Test A special type of tinference

Matched Pairs – two forms • Pair individuals by certain characteristics • Randomly select treatment for individual A • Individual B is assigned to other treatment • Assignment of B is dependent on assignment of A • Individual persons or items receive both treatments • Order of treatments are randomly assigned or before & after measurements are taken • The two measures are dependent on the individual

Is this an example of matched pairs? 1)A college wants to see if there’s a difference in time it took last year’s class to find a job after graduation and the time it took the class from five years ago to find work after graduation. Researchers take a random sample from both classes and measure the number of days between graduation and first day of employment No, there is no pairing of individuals, you have two independent samples

Is this an example of matched pairs? 2) In a taste test, a researcher asks people in a random sample to taste a certain brand of spring water and rate it. Another random sample of people is asked to taste a different brand of water and rate it. The researcher wants to compare these samples No, there is no pairing of individuals, you have two independent samples – If you would have the same people taste both brands in random order, then it would bean example of matched pairs.

Is this an example of matched pairs? 3) A pharmaceutical company wants to test its new weight-loss drug. Before giving the drug to a random sample, company researchers take a weight measurement on each person. After a month of using the drug, each person’s weight is measured again. Yes, you have two measurements that are dependent on each individual.

A whale-watching company noticed that many customers wanted to know whether it was better to book an excursion in the morning or the afternoon. You mayquestion, subtract either To test this the company collected the way – just be on careful following data 15 when randomly selected days over writing Ha the past month. (Note: days were not consecutive. ) Day 1 2 3 Morning 8 9 7 9 Afternoon 8 10 9 8 4 5 6 7 8 9 10 11 12 13 14 15 10 13 10 8 2 5 7 7 6 8 7 9 10 4 7 8 9 6 6 9 11 8 Since you have two values for each day, they are dependent on the day – making this data matched pairs First, you must find the differences for each day.

Day 1 2 3 4 Morning 8 9 7 9 10 13 10 Afternoon 8 10 9 8 9 Differenc es 0 -1 -2 1 1 Assumptions: 5 6 7 8 9 10 11 12 13 14 15 8 2 5 7 7 6 8 7 11 8 10 4 7 8 9 6 6 9 I subtracted: 2 Morning 2 -2 -1 -2 0 2 -2 – afternoon You could subtract the other way! • Have an SRS of days for whale-watching You need to state assumptions using the • s unknown differences! • Since the normal probability plot is approximately linear, the distribution of difference is approximately normal. Notice the granularity in this plot, it is still displays a nice linear relationship!

Differenc es 0 -1 -2 1 1 2 2 -2 -1 -2 0 2 -2 Is there sufficient evidence that more whales are sighted in the afternoon? H 0: m. D = 0 Ha : m. D < 0 Be careful writing your Ha! Think about how –you subtracted: If you subtract afternoon morning; then M-A Hma. D: m >0 Notice we for differences Dis If used afternoon more should the & it equals 0 since the null be should differences + or be -? that there is NO difference. Don’t look at numbers!!!! Where m. D is the true mean difference in whale sightings from morning minus afternoon

Differenc es 0 -1 -2 1 1 2 finishing the hypothesis test: 2 -2 -1 -2 0 2 -2 In your calculator, perform Notice thata ift-test you using the differences subtracted A-M, (L 3) test then your statistic t = +. 945, but pvalue would be the same Since p-value > a, I fail to reject H 0. There is insufficient evidence to suggest that more whales are sighted in the afternoon than in the morning.