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Ch10_Hyp_Testing

Hypothesis Tests Regarding a Parameter

QuestionAnswer
What is a “hypothesis”? A “hypothesis” is a statement regarding a characteristic of one or more populations.
In statistics, what do we mean by “hypothesis testing”? “Hypothesis testing” is a procedure, based on sample evidence and probability used to test statements regarding a characteristic of one or more populations.
How is the “null hypothesis” used in hypothesis testing? The “null hypothesis”, denoted H0, is a statement to be tested regarding a population parameter (statement of no change, no effect or no difference and is assumed true until evidence indicates otherwise.
How is the “alternative hypothesis” used in hypothesis testing? The “alternative hypothesis”, denoted H1, is a statement that we are trying to find evidence to support regarding the value of a population parameter.
Suppose the hypothesis test regarding a population parameter is set up as follows: H0: parameter = some value H1: parameter ≠ some value What type of hypothesis test is this: left-tailed, right-tailed or two-tailed? This is a Two-tailed hypothesis test.
Suppose the hypothesis test regarding a population parameter is set up as follows: H0: parameter = some value H1: parameter < some value What type of hypothesis test is this: left-tailed, right-tailed or two-tailed? This is a Left-tailed hypothesis test.
Suppose the hypothesis test regarding a population parameter is set up as follows: H0: parameter = some value H1: parameter > some value What type of hypothesis test is this: left-tailed, right-tailed or two-tailed? This is a Right-tailed hypothesis test.
According to one public official, the average length of a 911-call in 2008 was 3.50 minutes. A researcher believes that the average length of a 911-call has increased since then. State the Null and Alternative hypothesis. H0: µ = 3.5 H1: µ > 3.5 Right-tailed test
A part produced using an old machine with a standard deviation of the diameter of 0.052 inches. The engineering department believes that the new machine can reduce the standard deviation of the part. State the Null and Alternative hypothesis. H0: σ = 0.052 H1: σ < 0.052 Left-tailed test
John makes the statement that 38% of American adults exercise on a regular basis. A researcher believes that the actual percentage is different. State the Null and Alternative hypothesis. H0: p = 0.38 H1: p ≠ 0.38 Two-tailed test
In statistics, what do we mean by a “Type I error”? A Type I error is made when we reject the null hypothesis (H0) when the null hypothesis (H0) is true.
What symbol is used for the probability of making a Type I error? The symbol used for the probability of making a Type I error is α (“alpha”).
In statistics, what do we mean by a “Type II error”? A Type II error is made when we do not reject the null hypothesis (H0) when the alternative hypothesis (H1) is true.
What symbol is used for the probability of making a Type II error? The symbol used for the probability of making a Type II error is β (“beta”).
In statistics, how is the term “level of significance used? The “level of significance”, a, is the probability of making a Type I error.
What is the relationship between the probability of a Type I error and the probability of a Type II error? As the probability of a Type I error increases, the probability of a Type II error decreases, and vice-versa.
Is it proper to say that we “accept” the Null Hypothesis? No. We never “accept” the null hypothesis. Rather, we say that we “do not reject” the null hypothesis. This is just like the court system. We never declare a defendant “innocent”, but rather say the defendant is “not guilty”.
Suppose a researcher wants to test the claim that the average amount of the over-limit fee on credit cards in the U.S. is $39 and sets up the following hypotheses: H0: µ = 39 H1: µ > 39. Explain what it would mean to make a Type I error. A “Type I” error is made if the researcher concludes that the average amount of the over-limit fee on credit cards in the U.S. greater than $39 when the true average amount of the over-limit fee on credit cards in the U.S. is $39.
Suppose a researcher wants to test the claim that the average amount of the over-limit fee on credit cards in the U.S. is $39 and sets up the following hypotheses: H0: µ = 39 H1: µ > 39. Explain what it would mean to make a Type II error. A “Type II” error is made if the researcher concludes that the average amount of the over-limit fee on credit cards in the U.S. is $39 when the true average amount of the over-limit fee on credit cards in the U.S. is greater than $39.
Before testing hypotheses regarding a population mean assuming the population standard deviation is known, what requirements must be met? The following requirements must first be satisfied: 1. A simple random sample is obtained. 2. The population from which the sample is drawn is normally distributed, OR 3. The sample size is large (n≥30).
In hypothesis testing regarding a population mean assuming the population standard deviation is known, what is the formula for calculating the test statistic? Because the population standard deviation is “known”, we would use a “Z-Interval”. The formula for the test statistic would be: z0 = (x-bar – µ0)/(σ/sqrt(n))
Using the classical method of hypothesis testing regarding a population mean, assuming the population standard deviation is known, for a “two-tailed test”, how are critical values calculated? Since σ is known we are conducting a “Z-Test” with two-tails; so there will be two critical values: Using the TI-83/84 we have: z(α/2)= invNorm(1 – α/2, 0, 1) – z(α/2) = invNorm(α/2, 0, 1).
How are the critical z-values, z(α/2) and – z(α/2), used to test the hypotheses for a two-tailed test? For a two-tailed test, the Null hypothesis is REJECTED if the test statistic (z0) is in the either tail (the CRITICAL REGIONS). That is, if z0 < – z(α/2) OR z0 > z(α/2). Otherwise, we DO NOT REJECT the Null hypothesis.
Using the classical method of hypothesis testing regarding a population mean, assuming the population standard deviation is known, for a “left-tailed test”, how is the critical value calculated? Assuming the population standard deviation is known, we are conducting a “Z-Test”. Since this is a left-tailed test, there will be one critical value: – z(α). For the TI-83/84, use: invNorm(α, 0, 1) to find the critical Z value, –z(α).
How is the critical z-value,– z(α), used to test the hypotheses for a left -tailed test? For a left-tailed test, the Null hypothesis is REJECTED if the test statistic (z0) is in the left tail (the CRITICAL REGION). That is, if z0 < – z(α). Otherwise, we DO NOT REJECT the Null hypothesis.
Using the classical method of hypothesis testing regarding a population mean, assuming the population standard deviation is known, for a “Right-tailed test”, how is the critical value calculated? Assuming the population standard deviation is known, we are conducting a “Z-Test”. Since this is a right-tailed test, there will be one critical value: z(α). For the TI-83/84, use: invNorm(1 – α, 0, 1) to find the critical Z value, z(α).
How is the critical z-value, z(α), used to test the hypotheses for a right -tailed test? For a right-tailed test, the Null hypothesis is REJECTED if the test statistic (z0) is in the right tail (the CRITICAL REGION). That is, if z0 > z(α). Otherwise, we DO NOT REJECT the Null hypothesis.
What does the “p-value” represent when you are testing a hypothesis concerning a population mean, µ? When a hypothesis concerning a population mean, µ, is tested, the “p-value” is the probability of obtaining a sample mean at least as extreme as the value of “x-bar” observed if the value assumed by the null hypothesis were true.
Under the p-value approach, how is the p-value used to test the hypotheses? Once the p-value is calculated, the decision is straightforward: 1) If the p-value < α, REJECT the Null hypothesis; 2) If the p-value ≥ α, DO NOT REJECT the Null Hypotheses.
Suppose you are performing a left-tailed test, where the population standard deviation is known, and the test statistic is -1.25. What is the p-value? We are using a Z-test because the population standard deviation is known. Since this is a left-tailed test, we are looking for the p-value = P(Z < -1.25). Using the TI-83/84, we have: normalcdf(-1E99, -1.25, 0, 1) = 0.1056
In hypothesis testing regarding a population mean assuming the population standard deviation is “unknown”, what is the formula for calculating the test statistic? Because the population standard deviation is “unknown”, we would use a “T-Interval”. The formula for the test statistic would be: t0 = (x-bar – µ0)/(s/sqrt(n))
To test hypotheses regarding the population mean with s “unknown”, what requirements must be checked before performing the test? The following must be checked first: 1. simple random sampling obtained. 2. The sample has no outliers, and the population is normally distributed, OR 3. The sample size is large (n≥30). Same as for a Z-test for a population mean.
Using the classical method of hypothesis testing regarding a population mean, assuming the population standard deviation is “unknown”, for a two-tailed test, how are critical values calculated? Since σ is “unknown”, we are conducting a “T-Test”, two-tailed test so there are two critical values with n – 1 degrees of freedom. For the TI-83/84, use: t(α/2) = invT(1 – α/2, n – 1. and – t(α/2) = invT(α/2, n – 1)
How are the critical t-values, t(α/2) and – t(α/2), used to test the hypotheses for a two-tailed test? For a two-tailed test, the Null hypothesis is REJECTED if the test statistic (t0) is in the either tail (the CRITICAL REGIONS). That is, if t0 < – t(α/2) OR t0 > t(α/2), for n – 1 degrees of freedom. Otherwise, we DO NOT REJECT the Null hypothesis.
Using the classical method of hypothesis testing regarding a population mean, assuming the population standard deviation is “unknown”, for a left-tailed test, how is the critical value calculated? Since σ is “unknown”, we are conducting a “T-Test”. Since this is a left-tailed test, there will be one critical value with n – 1 degrees of freedom: For the TI-83/84, use: – t(α) = invT(α, n – 1) to find the critical T value.
How is the critical t-value,– t(α), used to test the hypotheses for a left -tailed test? For a left-tailed test, the Null hypothesis is REJECTED if the test statistic (t0) is in the left tail (the CRITICAL REGION). That is, if t0 < – t(α) for n – 1 degrees of freedom. Otherwise, we DO NOT REJECT the Null hypothesis.
Using the classical method of hypothesis testing regarding a population mean, assuming the population standard deviation is “unknown”, for a Right-tailed test, how is the critical value calculated? Since σ is “unknown”, we are conducting a “T-Test”. Since this is a “right-tailed test”, there will be one critical value with n – 1 degrees of freedom. For the TI-83/84, use: t(α) = invT(1 – α, n – 1).
How is the critical t-value, t(α), used to test the hypotheses for a right tailed test? For a right-tailed test, the Null hypothesis is REJECTED if the test statistic (t0) is in the right tail (the CRITICAL REGION). That is, if t0 > t(α) for n – 1 degrees of freedom. Otherwise, we DO NOT REJECT the Null hypothesis.
For the T-test, under the p-value approach, how is the p-value used to test the hypotheses? Once the p-value is calculated, the decision is straightforward: 1) If the p-value < α, REJECT the Null hypothesis; 2) If the p-value ≥ α, DO NOT REJECT the Null Hypotheses.
Suppose you are performing a left-tailed test, where the population standard deviation is “unknown”, and the test statistic is -1.25 and the sample size is n = 20. What is the p-value? We are using a T-test because the population standard deviation is “unknown”. Since this is a left-tailed test, we are looking for the p-value = P(T < -1.25). Using the TI-83/84, we have: tcdf(-1E99, -1.25, 19) = 0.1132
Before performing a hypothesis test for a “population proportion”, p, what requirements must be met? We must first check the following: 1. The sample is obtained by simple random sampling. 2. The sampled values are independent of each other, i.e., n ≤ 0.05N. 3. np0(1-p0) ≥ 10.
How would the hypothesis test for a claimed population proportion, p, be performed? The steps would be the same as those used for testing a hypothesis for a population mean, using the Z-interval. Either the “classical approach” or “p-value approach” may be used. However, the formula for the calculating the test-statistic is different.
What formula is used for calculating the test-statistic when we are testing a hypothesis about population proportion? The formula used for calculating the test-statistic when we are testing a hypothesis about population proportion is: Z0 = (p-hat – p0)/sqrt[p0(1 – p0)/n]
A researcher claims that 48% of Americans are unhappy with politicians in Washington; in a poll 535 out of 1010 adults stated they are not happy with politicians in Washington. Calculate the value of the test statistic to be used in the hypothesis test. First find P-hat = 535/1010 = 0.5297. Therefore, the test statistic is: Z0 = (0.5297 – 0.48)/sqrt[0.48(1 – 048)/1010] = 3.162
Based on the previous problem, if you want to test whether the percentage of Americans unhappy with politicians in Washington is more than the researcher claimed, using the p-value method, what conclusion would you reach? Use α= 0.05. First, find p-value = P(Z > 3.162) = 0.000784; p-value (0.000784) < α = 0.05; REJECT H0. There is sufficient evidence to conclude the percentage of Americans unhappy with politicians in Washington is more than the 48% claimed by the researcher.
Created by: wgriffin410