In what circumstances is the t statistic used instead of a z-score for a hypothesis test?

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Statistics for the Behavorial Sciences 9th Edition Chap. 9
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Why is the t statistic used instead of the Z statistic?
Under what circumstance is't statistic used instead of z
In what circumstances is the t
What is the difference between a t statistic and a Z statistic?

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Statistics for the Behavorial Sciences 9th Edition Chap. 9

Question	Answer
(Sm) used as an estimate of the real standard error (Om), when the value of O is unknown. Computed from the sample variance or sample standard deviation and provides an estimate of the standard distance between a sample mean M and the population mean U.	estimated standard error
used to test hypotheses about an unknown population mean, U, when the value of O is unknown.	t statistic
describes the number of scores in a sample that are independent and free to vary. (n -1)	degrees of freedom
the complete set of t values computed for every possible random sample size (n) or a specific degrees of freedom (df). Approximates the shape of normal distribution.	t distribution
Under what circumstances is a t statistic used instead of a z-score for a hypothesis test?	A t statistic is used instead of a z-score when the population standard deviation and variance are not known.
A sample of n=9 scores has SS = 288. Compute the variance for the sample.	36
A sample of n=9 scores has SS = 288. Compute the estimated standard error for the sample mean.	2
True or False. In general a distribution of t statistics is flatter and more spread out than the standard normal distribution.	True - As sample size and df increase, the variability in the t distribution decreases, and more closely resembles a normal distribution.
A researcher reports a t statistic with df = 20. How many individuals participated in the study.	n = 21
For df=15, find the value(s) of t associated with the top 5% of the distribution.	+1.753
For df=15, find the value(s) of t associated with the middle 95% of the distribution.	+-2.131
For df=15, find the value(s) of t associated with the middle 99% of the distribution.	+-2.947
Sample = n=4, U=40 Treatment sample = M=44, variance s2=16. Is this sample sufficient to conclude that the treatment has a significant effect?	No - Fail to reject H0, treatment does not have a significant effect
Sample = n=4, U=40 Treatment sample = M=44, variance s2=16. If all factors remained constant and sample size increased to n = 16, is sample sufficient to prove significant effect?	Yes, Reject NO. Treatment has significant effect.
an interval or range of values, centered around a sample statistic.	confidence interval
If all other factors are held constant, an 80% confidence interval is wider than a 90% confidence interval. (True or False?)	False - Greater confidence requires wider interval.
If all other factors are held constant, a confidence interval computed from a sample of n=25 is wider than a confidence interal computed from a sample of n = 100.	True. The smaller sample produces a wider interval.

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35) A manufacturer makes ball bearings that are supposed to have a mean weight of 30 g. A 35) retailer suspects that the mean weight is actually less than 30 g. The mean weight for a random sample of 16 ball bearings is 28.6 g with a standard deviation of 4.2 g. At the 0.05 significance level, test the claim that the sample comes from a population with a mean weight less than 30 g. Use the traditional method of testing hypotheses. 36) A cereal company claims that the mean weight of the cereal in its packets is 14 oz. The 36) weights (in ounces) of the cereal in a random sample of 8 of its cereal packets are listed below. 14.6 13.8 14.1 13.7 14.0 14.4 13.6 14.2 Test the claim at the 0.01 significance level. Use the P-value method to test the given hypothesis. Assume that the samples are independent and that they have been randomly selected. 37) A marketing survey involves product recognition in New York and California. Of 558 37) New Yorkers surveyed, 193 knew the product while 196 out of 614 Californians knew the product. At the 0.05 significance level, test the claim that the recognition rates are the same in both states. 38) In a random sample of 360 women, 65% favored stricter gun control laws. In a random 38) sample of 220 men, 60% favored stricter gun control laws. Test the claim that the proportion of women favoring stricter gun control is higher than the proportion of men favoring stricter gun control. Use a significance level of 0.05. 39) Seven of 8500 people vaccinated against a certain disease later developed the disease. 18 39) of 10,000 people vaccinated with a placebo later developed the disease. Test the claim that the vaccine is effective in lowering the incidence of the disease. Use a significance level of 0.02. Construct the indicated confidence interval for the difference between population proportions p1 - p2. Assume that the samples are independent and that they have been randomly selected. 40) A marketing survey involves product recognition in New York and California. Of 558 40) New Yorkers surveyed, 193 knew the product while 196 out of 614 Californians knew the product. Construct a 99% confidence interval for the difference between the two population proportions. 41) In a random sample of 500 people aged 20-24, 22% were smokers. In a random sample of 41) 450 people aged 25-29, 14% were smokers. Construct a 95% confidence interval for the difference between the population proportions p1 - p2. 5 Test the indicated claim about the means of two populations. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. Use the P-value method as indicated. 42) A researcher wishes to determine whether people with high blood pressure can reduce 42) their blood pressure, measured in mm Hg, by following a particular diet. Use a significance level of 0.01 to test the claim that the treatment group is from a population with a smaller mean than the control group. Use the P-value method of hypothesis testing. Treatment Group Control Group n1 = 35 n2 = 28 x1 = 189.1 x2 = 203.7 s1 = 38.7 s2 = 39.2 43) A researcher was interested in comparing the GPAs of students at two different colleges. 43) Independent random samples of 8 students from college A and 13 students from college B yielded the following GPAs: College A College B 3.7 3.8 2.8 3.2 3.2 4.0 3.0 3.0 3.6 2.5 3.9 2.6 2.7 3.8 4.0 3.6 2.5 3.6 2.8 3.9 3.4 Use a 0.10 significance level to test the claim that the mean GPA of students at college A is different from the mean GPA of students at college B. Use the P-value method of hypothesis testing. (Note: x1 = 3.1125, x2 = 3.4385, s1 = 0.4357, s2 = 0.5485.) Construct the indicated confidence interval for the difference between the two population means. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. 44) A researcher was interested in comparing the amount of time spent watching television 44) by women and by men. Independent simple random samples of 14 women and 17 men were selected, and each person was asked how many hours he or she had watched television during the previous week. The summary statistics are as follows. Women Men x1 = 12.3 hrs x2 = 13.7 hrs s1 = 3.9 hrs s2 = 5.2 hrs n1 = 14 n2 = 17 Construct a 99% confidence interval for µ1 - µ2, the difference between the mean amount of time spent watching television for women and the mean amount of time spent watching television for men. 6 45) A researcher was interested in comparing the resting pulse rates of people who exercise 45) regularly and the pulse rates of people who do not exercise regularly. She obtained independent simple random samples of 16 people who do not exercise regularly and 12 people who do exercise regularly. The resting pulse rates (in beats per minute) were recorded and the summary statistics are as follows. Do not exercise regularly Exercise regularly x1 = 73.3 beats/min x2 = 68.0 beats/min s1 = 10.9 beats/min s2 = 8.2 beats/min n1 = 16 n2 = 12 Construct a 95% confidence interval for µ1 - µ2, the difference between the mean pulse rate of people who do not exercise regularly and the mean pulse rate of people who exercise regularly. Construct a confidence interval for µd, the mean of the differences d for the population of paired data. Assume that the population of paired differences is normally distributed. 46) A coach uses a new technique in training middle distance runners. The times for 9 46) different athletes to run 800 meters before and after this training are shown below. Athlete A B C D E F G H I Time before training (seconds) 115.2 120.9 108.0 112.4 107.5 119.1 121.3 110.8 122.3 Time after training (seconds) 116.0 119.1 105.1 111.9 109.1 115.2 118.5 110.7 120.9 Construct a 99% confidence interval for the mean difference of the "before" minus "after" times. Use the P-value method of hypothesis testing to test the given claim about the means of two populations. Assume that two dependent samples have been randomly selected from normally distributed populations. 47) Five students took a math test before and after tutoring. Their scores were as follows. 47) Subject A B C D E Before 71 66 78 78 72 After 75 75 76 81 84 Using a 0.01 level of significance, test the claim that the tutoring has an effect on the math scores. 48) Ten different families are tested for the number of gallons of water a day they use before 48) and after viewing a conservation video. At the 0.05 significance level, test the claim that the mean is the same before and after the viewing. Before 33 33 38 33 35 35 40 40 40 31 After 34 28 25 28 35 33 31 28 35 33 Find the value of the linear correlation coefficient r. 49) x 38.5 13.8 48.6 41.8 25.7 y 3 9 7 10 8 49) 50) The paired data below consist of the temperatures on randomly chosen days and the 50) amount a certain kind of plant grew (in millimeters): Temp 62 76 50 51 71 46 51 44 79 Growth 36 39 50 13 33 33 17 6 16 7 Use the given data to find the best predicted value of the response variable. 51) ^ Ten pairs of data yield r = 0.003 and the regression equation y = 2 + 3x. Also, y = 5.0. What is the best predicted value of y for x = 2? 51) Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary. 52) x 0 3 4 5 12 y 8 2 6 9 12 52) 53) Two different tests are designed to measure employee productivity and dexterity. Several 53) employees are randomly selected and tested with these results. Productivity Dexterity 23 25 28 21 21 25 26 30 34 36 49 53 59 42 47 53 55 63 67 75 Perform the indicated goodness-of-fit test. 54) A company manager wishes to test a union leader's claim that absences occur on the 54) different week days with the same frequencies. Test this claim at the 0.05 level of significance if the following sample data have been compiled. Day Mon Tue Wed Thur Fri Absences 37 15 12 23 43 55) Use a significance level of 0.01 to test the claim that workplace accidents are distributed 55) on workdays as follows: Monday 25%, Tuesday: 15%, Wednesday: 15%, Thursday: 15%, and Friday: 30%. In a study of 100 workplace accidents, 27 occurred on a Monday, 16 occurred on a Tuesday, 17 occurred on a Wednesday, 14 occurred on a Thursday, and 26 occurred on a Friday. Use a Λ 2 test to test the claim that in the given contingency table, the row variable and the column variable are independent. 56) Tests for adverse reactions to a new drug yielded the results given in the table. At the 0.05 56) significance level, test the claim that the treatment (drug or placebo) is independent of the reaction (whether or not headaches were experienced). Drug Placebo Headaches 11 7 No headaches 73 91 57) The table below shows the age and favorite type of music of 668 randomly selected 57) people. Rock Pop Classical 15-25 50 85 73 25-35 68 91 60 35-45 90 74 77 Use a 5 percent level of significance to test the null hypothesis that age and preferred music type are independen

Why is the t statistic used instead of the Z statistic?

The major difference between using a Z score and a T statistic is that you have to estimate the population standard deviation. The T test is also used if you have a small sample size (less than 30).

Under what circumstance is't statistic used instead of z

Under what circumstances is a T Statistic used instead of a z-score for a hypothesis test? When the population standard deviation and variance are not known.