"If you are wearing a hat, you are a baseball fan." and "If you are not a baseball fan, you are not wearing a hat." It's called a [B]contrapositive[/B]. definition: : a proposition or theorem formed by contradicting both the subject and predicate or both the hypothesis and conclusion of a given proposition

A candidate for mayor wants to gauge potential voter reaction to an increase recreational services by estimating the proportion of voter who now use city services. If we assume that 50% of the voters require city recreational services, what is the probability that 40% or fewer voters in a sample of 100 actually will use these city services? First, let's do a test on the proportion using our [URL='http://www.mathcelebrity.com/proportion_hypothesis.php?x=+40&n=+100&ptype=%3D&p=+0.5&alpha=+0.05&pl=Proportion+Hypothesis+Testing']proportion hypothesis calculator[/URL]: We get Z = -2 Now use the [URL='http://www.mathcelebrity.com/zscore.php?z=p%28z%3C-2%29&pl=Calculate+Probability']z-score calculator[/URL] to get P(z<-2) = [B]0.02275[/B]

A firm wants to know with a 98% level of confidence if it can claim that the boxes of detergent it sells contain more than 500g of detergent. From past experience the firm knows that the amount of detergent in the boxes is normally distributed. The firm takes a random sample of n =25 and finds that X = 520 g and s = 75g. What's your final conclusion? (Ho: u = 500; Ha: u > 500) [URL='http://www.mathcelebrity.com/mean_hypothesis.php?xbar=520&n=25&stdev=75&ptype==&mean=500&alpha=0.02&pl=Mean+Hypothesis+Testing']Perform a hypothesis testing of the mean[/URL] [B]Yes, accept null hypothesis[/B]

A Government antipollution spokeperson asserts that more than 80% of the plants in the Glassboro area meet the antipollution standards. An antipollution advocate does not believe the government claim. She takes a random sample of published data on pollution emission for 64 plants in the area and finds that 56 plants meet the pollution standards. Do the sample data support the government claim at the 1% level of significance? (H0: ρ=0.8; H_a: ρ>0.8) [URL='http://www.mathcelebrity.com/proportion_hypothesis.php?x=+56&n=64&ptype==&p=+0.8&alpha=+0.01&pl=Proportion+Hypothesis+Testing']Perform a hypothesis testing of a proportion[/URL] [B]Accept null hypothesis[/B]

The mean credit card debt among households in one state is $8400. A hypothesis test is to be performed to decide whether the mean credit card debt for households in the formerly affluent town of Rich-No-More differs from the mean credit card debt for the state.

A professor assumed there was a correlation between the amount of hours people were expose to sunlight and their blood vitamin D level. The null hypothesis was that the population correlation was__ a. Positive 1.0 b. Negative 1.0 c. Zero d. Positive 0.50 [B]c. Zero[/B] Reason: Since the professor wanted to assume a correlation (either positive = 1.0 or negative = -1.0), then we take the other side of that assumption for our null hypothesis and say that there is no correlation (Zero)

A researcher believed that there was a difference in the amount of time boys and girls at 7th grade studied by using a two-tailed t test. Which of the following is the null hypothesis? a. Mean of hours that boys studied per day was equal to mean of hours that girls studied per day b. Mean of hours that boys studied per day was greater than mean of hours that girls studied per day c. Mean of hours that boys studied per day was smaller than mean of hours that girls studied per day d. Mean of hours that boys studied per day was smaller than or equal to mean of hours that girls studied per day [B]a. Mean of hours that boys studied per day was equal to mean of hours that girls studied per day[/B] Reason is that in hypothesis testing, you take a position other than what is assumed or what is being tested as the null hypothesis

A researcher posed a null hypothesis that there was no significant difference between boys and girls on a standard memory test. He randomly sampled 100 girls and 120 boys in a community and gave them the standard memory test. The mean score for girls was 70 and the standard deviation of mean was 5.0. The mean score for boys was 65 and the standard deviation of mean was 6.0. What's the absolute value of the difference between means? |70 -65| = |5| = 5

A researcher posed a null hypothesis that there was no significant difference between boys and girls on a standard memory test. He randomly sampled 100 girls and 100 boys in a community and gave them the standard memory test. The mean score for girls was 70 and the standard deviation of mean was 5.0. The mean score for boys was 65 and the standard deviation of mean was 5.0. What is the standard error of the difference in means? [B]0.707106781187[/B] using our [URL='http://www.mathcelebrity.com/meandiffconf.php?n1=+100&xbar1=70&stdev1=5&n2=+100&xbar2=65&stdev2=5&conf=+99&pl=Hypothesis+Test']difference of means calculator[/URL]

A researcher posed a null hypothesis that there was no significant difference between boys and girls on a standard memory test. He randomly sampled 100 girls and 100 boys in a community and gave them the standard memory test. The mean score for girls was 70 and the standard deviation of mean was 5.0. The mean score for boys was 65 and the standard deviation of mean was 5.0. What's the t-value (two-tailed) for the null hypothesis that boys and girls have the same test scores? t = 7.07106781187 using our [URL='http://www.mathcelebrity.com/meandiffconf.php?n1=+100&xbar1=70&stdev1=5&n2=+100&xbar2=65&stdev2=5&conf=+99&pl=Hypothesis+Test']difference of means calculator[/URL]

A student hypothesized that girls in his class had the same blood pressure levels as boys. The probability value for his null hypothesis was 0.15. So he concluded that the blood pressures of the girls were higher than boys'. Which kind of mistake did he make? a. Type I error b. Type II error c. Type I and Type II error d. He did not make any mistake [B]d. He did not make any mistake[/B] [I]p value is high, especially using a significance level of 0.05[/I]

A student posed a null hypothesis that during the month of September, the mean daily temperature of Boston was the same as the mean daily temperature of New York. His alternative hypothesis was that mean temperatures in these two cities were different. He found the P value of his null hypothesis was 0.56. Thus, he could conclude: a. In September, Boston was colder than New York b. In September, Boston was warmer than New York c. He may reject the null hypothesis d. He failed to reject the null hypothesis [B]d. He failed to reject the null hypothesis[/B] [I]Higher p value tells us we cannot reject the null hypothesis[/I]

A teacher hypothesized that in her class, grades of girls on a chemistry test were the same as grades of boys. If the probability value of her null hypothesis was 0.56, it suggested: a. We failed to reject the null hypothesis b. Boys' grades were higher than girls' grades c. Girls' grades were higher than boys' grades d. The null hypothesis was rejected [B]a. We failed to reject the null hypothesis[/B] Due to a high probability.

As the sample size increases, we assume: a. α increases b. β increases c. The probability of rejecting a hypothesis increases d. Power increases [B]d. Power increases[/B] [LIST] [*]Power increases if the standard deviation is smaller. [*]If the difference between the means is bigger, the power is bigger. [*]Sample size also increases power [/LIST]

Free Confidence Interval/Hypothesis Testing for the Difference of Means Calculator - Given two large or two small distriutions, this will determine a (90-99)% estimation of confidence interval for the difference of means for small or large sample populations.
Also performs hypothesis testing including standard error calculation.

Conventionally, the null hypothesis is false if the probability value is: a. Greater than 0.05 b. Less than 0.05 c. Greater than 0.95 d. Less than 0.95 [B]b. Less than 0.05[/B] This is standard in hypothesis testing using p = 0.05

Free Difference of Proportions Test Calculator - Calculates a test statistic and conclusion for a hypothesis for the difference of proportions

[URL]http://www.mathcelebrity.com/mean_hypothesis.php?xbar=3.7&n=3.2&stdev=1.8&ptype=%3D&mean=4.2&alpha=0.05&pl=Mean+Hypothesis+Testing[/URL]

[URL]http://www.mathcelebrity.com/mean_hypothesis.php?xbar=20.5&n=11&stdev=7&ptype=%3D&mean=18.7&alpha=0.01&pl=Mean+Hypothesis+Testing[/URL]

[URL]http://www.mathcelebrity.com/mean_hypothesis.php?xbar=469&n=40&stdev=73&ptype=%3D&mean=475&alpha=0.10&pl=Mean+Hypothesis+Testing[/URL]

A sample mean, sample size, and population standard deviation are given. Use the one-mean z-test to perform the required hypothesis test about the mean, µ, of the population from which the sample was drawn x = 3.26 , S = 0.55, σN= 9, H0: µ = 2.85, Ha: µ > 2.85 , α = 0.01

sample mean, sample size, and population standard deviation are given. Use the one-mean z-test to perform the required hypothesis test at the given significance level. x = 3.7, n = 32, σ = 1.8, H0: µ = 4.2 , Ha: µ ≠ 4.2 , α = 0.05

A sample mean, sample size, and population standard deviation are given. Use the one-mean z-test to perform the required hypothesis test at the given significance level. x = 20.5, n = 11, σ = 7, H0: µ = 18.7 , Ha: µ ≠ 18.7 , α = 0.01

Perform a one-sample z-test for a population mean. Be sure to state the hypotheses and the significance level, to compute the value of the test statistic, to obtain the P-value, and to state your conclusion. Five years ago, the average math SAT score for students at one school was 475. A teacher wants to perform a hypothesis test to determine whether the mean math SAT score of students at the school has changed. The mean math SAT score for a random sample of 40 students from this school is 469. Do the data provide sufficient evidence to conclude that the mean math SAT score for students at the school has changed from the previous mean of 475? Perform the appropriate hypothesis test using a significance level of 10%. Assume that Η = 73.

Free Hypothesis Testing for a proportion Calculator - Performs hypothesis testing using a test statistic for a proportion value.

Free Hypothesis testing for the mean Calculator - Performs hypothesis testing on the mean both one-tailed and two-tailed and derives a rejection region and conclusion

If the P-value of a hypothesis test is 0.40, you conclude a. You accept the null hypothesis b. You reject the null hypothesis c. You failed to reject the null hypothesis d. You think there is a significant difference [B]c. You failed to reject the null hypothesis[/B] [I]due to a high p value, especially above 0.05[/I]

If the probability that you will correctly reject a false null hypothesis is 0.80 at 0.05 significance level. Therefore, α is__ and β is__. [LIST] [*]α represents the significance level of 0.05 [*]We want the Power of a Test which is 1 - β = 0.8 so β = 0.20 [/LIST] Our answer is: [B]0.05, 0.20 [/B]

Imagine a researcher posed a null hypothesis that in a certain community, the average energy expenditure should be 2,100 calories per day. He randomly sampled 100 people in that community. After he computed the t value by calculating a two-tailed t-statistic, he found that the probability value was 0.10. Thus, he concluded: a. The average energy expenditure was bigger than 2,100 calories per day b. The average energy expenditure was smaller than 2,100 calories per day c. He could not reject the null hypothesis that the average energy expenditure was 2,100 calories per day d. The average energy expenditure was either more than 2,100 calories per day or less than 2,100 calories per day [B]c. He could not reject the null hypothesis that the average energy expenditure was 2,100 calories per day[/B] [I]p-value is higher than 0.05[/I]

Imagine that a researcher wanted to know the average weight of 5th grade boys in a high school. He randomly sampled 5 boys from that high school. Their weights were: 120 lbs., 99 lbs, 101 lbs, 87 lbs, 140 lbs. The researcher posed a null hypothesis that the average weight for boys in that high school should be 100 lbs. What is the [B][U]absolute value[/U][/B] of calculated t that we use for testing the null hypothesis? Mean is 109.4 and Standard Deviation = 20.79182532 using our [URL='http://www.mathcelebrity.com/statbasic.php?num1=120%2C99%2C101%2C87%2C140&num2=+0.2%2C0.4%2C0.6%2C0.8%2C0.9&pl=Number+Set+Basics']statistics calculator[/URL] Now use those values and calculate the t-value Abs(t value) = (100 - 109.4)/ 20.79182532/sqrt(5) Abs(tvalue) = [B]1.010928029[/B]

Free Mcnemar Test Calculator - Given a 2 x 2 contingency table and a significance level, this will determine the test statistic, critical value, and hypothesis conclusion using a Mcnemar test.

Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test, [U][B]the Type I error is[/B][/U]: a. to conclude that the current mean hours per week is higher than 4.5, when in fact, it is higher
b. to conclude that the current mean hours per week is higher than 4.5, when in fact, it is the same
c. to conclude that the mean hours per week currently is 4.5, when in fact, it is higher
d. to conclude that the mean hours per week currently is no higher than 4.5, when in fact, it is not higher [B]b. to conclude that the current mean hours per week is higher than 4.5, when in fact, it is the same [/B] [I]A Type I error is when you reject the null hypothesis when it is in fact true[/I]

Free Random Sampling from the Normal Distribution Calculator - This performs hypothesis testing on a sample mean with critical value on a sample mean or calculates a probability that Z <= z or Z >= z using a random sample from a normal distribution.

Free Random Test Calculator - Given a set of data and an α value, this determines the test statistic and accept/reject hypothesis based on randomness of a dataset.

Free Sign Test Calculator - This will determine whether to accept or reject a null hypothesis based on a number set, mean value, alternative hypothesis, and a significance level using the Sign Test.

Six Years ago, 12.2% of registered births were to teenage mothers. A sociologist believes that the percentage has decreased since then. (a) Which of the following is the hypothesis to be conducted? A. H0: p = 0.122, H1 p > 0.122 B. H0: p = 0.122, H1 p <> 0.122 C. H0: p = 0.122, H1 p < 0.122 (b) Which of the following is a Type I error? A. The sociologist rejects the hypothesis that the percentage of births to teenage mothers is 12.2%, when the true percentage is less than 12.2% B. The sociologist fails to reject the hypothesis that the percentage of births to teenage mothers is 12.2%, when the true percentage is less than 12.2% C. The sociologist rejects the hypothesis that the percentage of births to teenage mothers is 12.2%, when it is the true percentage. c) Which of the following is a Type II error? A. The sociologist rejects the hypothesis that the percentage of births to teenage mothers is 12.2%, when it is the true percentage B. The sociologist fails to reject the hypothesis that the percentage of births to teenage mothers is 12.2%, when it is the true percentage C. The sociologist fails to reject the hypothesis that the percentage of births to teenage mothers is 12.2%, when the true percentage is less than 12.2% (a) [B]C H0: p = 0.122, H1: p < 0.122[/B] because a null hypothesis should take the opposite of what is being assumed. So the assumption is that nothing has changed while the hypothesis is that the rate has decreased. (b) [B]C.[/B] The sociologist rejects the hypothesis that the percentage of births to teenage mothers is 12.2%, when it is the true percentage. Type I Error is rejecting the null hypothesis when it is true c) [B]C.[/B] The sociologist fails to reject the hypothesis that the percentage of births to teenage mothers is 12.2%, when the true percentage is less than 12.2% Type II Error is accepting the null hypothesis when it is false.

Suppose a firm producing light bulbs wants to know if it can claim that its light bulbs it produces last 1,000 burning hours (u). To do this, the firm takes a random sample of 100 bulbs and find its average life time (X=980 hrs) and the sample standard deviation s = 80 hrs. If the firm wants to conduct the test at the 1% of significance, what's you final suggestion? (i..e, Should the producer accept the Ho that its light bulbs have a 1,000 burning hrs. at the 1% level of significance?) Ho: u = 1,000 hours. Ha: u <> 1,000 hours. [URL='http://www.mathcelebrity.com/mean_hypothesis.php?xbar=+980&n=+100&stdev=+80&ptype==&mean=+1000&alpha=+0.01&pl=Mean+Hypothesis+Testing']Perform a hypothesis test of the mean[/URL] [B]Yes, accept null hypothesis[/B]

Suppose that the manager of the Commerce Bank at Glassboro determines that 40% of all depositors have a multiple accounts at the bank. If you, as a branch manager, select a random sample of 200 depositors, what is the probability that the sample proportion of depositors with multiple accounts is between 35% and 50%? [URL='http://www.mathcelebrity.com/proportion_hypothesis.php?x=50&n=+100&ptype==&p=+0.4&alpha=+0.05&pl=Proportion+Hypothesis+Testing']50% proportion probability[/URL]: z = 2.04124145232 [URL='http://www.mathcelebrity.com/proportion_hypothesis.php?x=+35&n=+100&ptype==&p=+0.4&alpha=+0.05&pl=Proportion+Hypothesis+Testing']35% proportion probability[/URL]: z = -1.02062072616 Now use the [URL='http://www.mathcelebrity.com/zscore.php?z=p%28-1.02062072616

The margarita is one of the most common tequila-based cocktails, made with tequila, triple sec, and lime juice, often served with salt on the glass rim. A manager at a local bar is concerned that the bartender is not using the correct amounts of the three ingredients in more than 50% of margaritas. He secretly observed the bartender and found that he used the CORRECT amounts in only 9 out of the 39 margaritas in the sample. Use the critical value approach to test if the manager's suspicion is justified at α = 0.10. Let p represent the proportion of all margaritas made by the bartender that have INCORRECT amounts of the three ingredients. Use Table 1. a. Select the null and the alternative hypotheses. [B]H0: p ≤ 0.50; HA: p > 0.50[/B] [B][/B] b. Calculate the sample proportion. (Round your answer to 3 decimal places.) 9/39 = [B]0.231 [/B] c. Calculate the value of test statistic. (Round your intermediate calculations to 4 decimal places and final answer to 2 decimal places.) Using our [URL='http://www.mathcelebrity.com/proportion_hypothesis.php?x=9&n=39&ptype=%3C&p=+0.5&alpha=+0.10&pl=Proportion+Hypothesis+Testing']proportion hypothesis calculator[/URL], we get: [B]Test Stat = -3.36[/B] [B][/B] d. Calculate the critical value. (Round your answer to 2 decimal places.) Using the link above, we get a critical value of [B]1.2816 [/B] e. What is the conclusion? [B]The manager’s suspicion is not justified since the value of the test statistic does not fall in the rejection region. Do not reject H0[/B] [B][/B]

The probability of failing to reject a false null hypothesis is ____ a. α b. 1 - α c. 1 - β d. β [B]d. β[/B]

True or False (a) The normal distribution curve is always symmetric to its mean. (b) If the variance from a data set is zero, then all the observations in this data set are identical. (c) P(A AND A^c)=1, where A^c is the complement of A. (d) In a hypothesis testing, if the p-value is less than the significance level α, we do not have sufficient evidence to reject the null hypothesis. (e) The volume of milk in a jug of milk is 128 oz. The value 128 is from a discrete data set. [B](a) True, it's a bell curve symmetric about the mean (b) True, variance measures how far a set of numbers is spread out. A variance of zero indicates that all the values are identical (c) True. P(A) is the probability of an event and P(Ac) is the complement of the event, or any event that is not A. So either A happens or it does not. It covers all possible events in a sample space. (d) False, we have sufficient evidence to reject H0. (e) False. Volume can be a decimal or fractional. There are multiple values between 127 and 128. So it's continuous.[/B]

When you conduct a hypothesis testing, at which of the following P-value, you feel more confident to reject the null hypothesis? a. 0.05 b. 0.01 c. 0.95 d. 0.03 [B]b. 0.01[/B] [I]The lower the p value, the more confident you are about rejecting the null hypothesis.[/I]

Which of the following descriptions of null hypothesis are correct? (Select all that apply) a. A null hypothesis is a hypothesis tested in significance testing. b. The parameter of a null hypothesis is commonly 0. c. The aim of all research is to prove the null hypothesis is true d. Researchers can reject the null hypothesis if the P-value is above 0.05 [B]a. A null hypothesis is a hypothesis tested in significance testing. [/B] [I]b. is false because a parameter can be anything we choose it to be c. is false because our aim is to disprove or fail to reject the null hypothesis d. is false since a p-value [U]below[/U] 0.05 is often the rejection level.[/I]

Which of the followings is the definition of power? a. Power is the probability of rejecting a null hypothesis b. Power is the probability of accepting a null hypothesis c. Power is the probability of accepting a false null hypothesis d. Power is the probability of rejecting a false null hypothesis [B]d. Power is the probability of rejecting a false null hypothesis[/B]

You choose an alpha level of .01 and then analyze your data. (a) What is the probability that you will make a Type I error given that the null hypothesis is true? (b) What is the probability that you will make a Type I error given that the null hypothesis is false. [B](a) 0.01. Instead, α is the probability of a Type I error given that the null hypothesis is true. If the null hypothesis is false, then it is impossible to make a Type I error.[/B] [B](b) Impossible Instead, α is the probability of a Type I error given that the null hypothesis is true. If the null hypothesis is false, then it is impossible to make a Type I error.[/B]

___is the probability of a Type II error; and ___ is the probability of correctly rejecting a false null hypothesis. a. 1 - β; β b. β; 1 - β; c. α; β; d. β; α [B]b. β; 1 - β;[/B] [LIST] [*]H0 is true = Correct Decision 1 - α Confidence Level = Size of a Test α = Type I Error [*]Ho is false = Type II Error β = Correct Decision 1 - β = Power of a Test [/LIST]