Hypothesis Tests Regarding a Parameter
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| What is a “hypothesis”? | show 🗑
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| show | “Hypothesis testing” is a procedure, based on sample evidence and probability used to test statements regarding a characteristic of one or more populations.
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| How is the “null hypothesis” used in hypothesis testing? | show 🗑
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| How is the “alternative hypothesis” used in hypothesis testing? | show 🗑
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| 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? | show 🗑
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| show | This is a Left-tailed hypothesis test.
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| show | This is a Right-tailed hypothesis test.
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| show | H0: µ = 3.5 H1: µ > 3.5 Right-tailed test
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| 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. | show 🗑
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| show | H0: p = 0.38 H1: p ≠ 0.38 Two-tailed test
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| show | A Type I error is made when we reject the null hypothesis (H0) when the null hypothesis (H0) is true.
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| What symbol is used for the probability of making a Type I error? | show 🗑
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| show | A Type II error is made when we do not reject the null hypothesis (H0) when the alternative hypothesis (H1) is true.
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| show | The symbol used for the probability of making a Type II error is β (“beta”).
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| In statistics, how is the term “level of significance used? | show 🗑
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| What is the relationship between the probability of a Type I error and the probability of a Type II error? | show 🗑
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| show | 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”.
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| show | 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.
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| show | 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.
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| show | 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).
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| In hypothesis testing regarding a population mean assuming the population standard deviation is known, what is the formula for calculating the test statistic? | show 🗑
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| show | 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).
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| show | 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.
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| show | 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).
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| show | 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.
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| 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? | show 🗑
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| How is the critical z-value, z(α), used to test the hypotheses for a right -tailed test? | show 🗑
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| show | 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.
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| show | 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.
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| 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? | show 🗑
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| show | 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))
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| show | 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.
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| show | 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)
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| How are the critical t-values, t(α/2) and – t(α/2), used to test the hypotheses for a two-tailed test? | show 🗑
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| show | 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.
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| How is the critical t-value,– t(α), used to test the hypotheses for a left -tailed test? | show 🗑
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| show | 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).
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| show | 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.
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| show | 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.
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| show | 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
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| show | 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.
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| show | 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.
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| show | 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]
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| show | 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
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| show | 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.
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