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EXAMPLE 1

EXAMPLE 1.

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EXAMPLE 1

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  1. EXAMPLE 1 • The processors of Fries’ Catsup indicate on the label that the bottle contains 16 ounces of catsup. The standard deviation of the process is 0.5 ounces. A sample of 36 bottles from last hour’s production revealed a mean weight of 16.12 ounces per bottle. At the .05 significance level is the process out of control? That is, can we conclude that the mean amount per bottle is different from 16 ounces?

  2. EXAMPLE 1 continued • Step 1: State the null and the alternative hypotheses: H0: MEW = 16; H1: MEW=/ 16 • Step 2: Select the level of significance. In this case we selected the .05 significance level. • Step 3: Identify the test statistic. Because we know the population standard deviation, the test statistic is z.

  3. EXAMPLE 1 continued • Step 4: State the decision rule: Reject H0 if z > 1.96or z < -1.96 • Step 5: Compute the value of the test statistic and arrive at a decision. Do not reject the null hypothesis. We cannot conclude the mean is different from 16 ounces.

  4. p-Value in Hypothesis Testing • A p-Value is the probability, assuming that the null hypothesis is true, of finding a value of the test statistic as extreme as or more than the observed value of the test. • If the p-Value is smaller than the significance level, H0 is rejected. • If the p-Value is larger than the significance level, H0 is not rejected.

  5. Computation of the p-Value • One-Tailed Test: p-Value = P{z absolute value of the computed test statistic value} • Two-Tailed Test: p-Value = 2P{z absolute value of the computed test statistic value} • From EXAMPLE 1, z = 1.44, and because it was a two-tailed test, the p-Value = 2P{z 1.44} = 2(.5-.4251) = .1498. Because .1498 > .05, do not reject H0.

  6. Testing for the Population Mean: Large Sample, Population Standard Deviation Unknown • Here SIGMA is unknown, so we estimate it with the sample standard deviation s. • As long as the sample size n> 30, z can be approximated with:

  7. EXAMPLE 2 • Roder’s Discount Store chain issues its own credit card. Lisa, the credit manager, wants to find out if the mean monthly unpaid balance is more than $400. The level of significance is set at .05. A random check of 172 unpaid balances revealed the sample mean to be $407 and the sample standard deviation to be $38. Should Lisa conclude that the population mean is greater than $400, or is it reasonable to assume that the difference of $7 ($407-$400) is due to chance?

  8. EXAMPLE 2 continued • Step 1:H0: µ <= $400, H1: µ > $400 • Step 2: The significance level is .05 • Step 3:Because the sample is large we can use the z distribution as the test statistic. • Step 4:H0 is rejected if z>1.65 • Step 5: Perform the calculations and make a decision. H0 is rejected. Lisa can conclude that the mean unpaid balance is greater than $400.

  9. Testing for a Population Mean: Small Sample, Population Standard Deviation Unknown • The test statistic is the t distribution. • The test statistic for the one sample case is given by:

  10. Example 3 The current rate for producing 5 amp fuses at Neary Electric Co. is 250 per hour. A new machine has been purchased and installed that, according to the supplier, will increase the production rate. A sample of 10 randomly selected hours from last month revealed the mean hourly production on the new machine was 256 units, with a sample standard deviation of 6 per hour. At the .05 significance level can Neary conclude that the new machine is faster?

  11. Example 3 continued • Step 1: State the null and the alternate hypothesis. H0: µ <= 250; H1: µ > 250 • Step 2: Select the level of significance. It is .05. • Step 3: Find a test statistic. It is the t distribution because the population standard deviation is not known and the sample size is less than 30.

  12. Example 3 continued • Step 4: State the decision rule. There are 10 – 1 = 9 degrees of freedom. The null hypothesis is rejected if t > 1.833. • Step 5: Make a decision and interpret the results. The null hypothesis is rejected. The mean number produced is more than 250 per hour.

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