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# Hypothesis Testing Calculations

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I need help, with the problems below.

1 Explain Type I and Type II errors. Use an example if needed.

2 Explain a one-tailed and two-tailed test. Use an example if
needed.

3 Define the following terms in your own words.
• Null hypothesis
• P-value
• Critical value
• Statistically significant

4 A homeowner is getting carpet installed. The installer is
charging her for 250 square feet. She thinks this is more than the actual space being carpeted. She asks a second installer to measure the space to confirm her doubt. Write the null hypothesis Ho and the alternative hypothesis Ha.

5 . Drug A is the usual treatment for depression in graduate
students. Pfizer has a new drug, Drug B, that it thinks may be more effective. You have been hired to design the test program. As part of your project briefing, you decide to explain the logic of statistical testing to the people who are going to be working for you.
• Write the research hypothesis and the null hypothesis.
• Then construct a table like the one below, displaying the outcomes that would constitute Type I and Type II error.
• Write a paragraph explaining which error would be more severe, and why.

6 . Cough-a-Lot children's cough syrup is supposed to contain 6
ounces of medicine per bottle. However since the filling machine is not always precise, there can be variation from bottle to bottle. The amounts in the bottles are normally distributed with σ = 0.3 ounces. A quality assurance inspector measures 10 bottles and finds the following (in ounces):
5.95 6.10 5.98 6.01 6.25 5.85 5.91 6.05 5.88 5.91
Are the results enough evidence to conclude that the bottles are not filled adequately at the labeled amount of 6 ounces per bottle?
a State the hypothesis you will test.
b Calculate the test statistic.
c Find the P-value.
d What is the conclusion?

7 Calculate a Z score when X = 20, μ = 17, and σ = 3.4.

8 Using a standard normal probabilities table, interpret the
results for the Z score in Problem 7.

9 Your babysitter claims that she is underpaid given the current
market. Her hourly wage is \$12 per hour. You do some research and discover that the average wage in your area is \$14 per hour with a standard deviation of 1.9. Calculate the Z score and use the table to find the standard normal probability. Based on your findings, should you give her a raise? Explain your reasoning as to why or why not.

10 Tutor O-Rama claims that their services will raise student
SAT math scores at least 50 points. The average score on the math portion of the SAT is μ=350 and σ=35. The 100 students who completed the tutoring program had an average score of 385 points.
a Is the students' average score of 385 points significant at
the 5% and 1% levels to support Tutor O-Rama's claim of at least a 50-point increase in the SAT score?

b. Is the Tutor O-Rama students' average score of 385 points significantly different at the 5% and 1% levels from the average score of 350 points on the math portion of the SAT? What conclusion can you make, based on your results, about the effectiveness of Tutor O-Rama's tutoring?

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#### Solution Preview

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Hypothesis
1. Explain Type I and Type II errors. Use an example if needed.
Type I error, also known as an "error of the first kind", an α error, or a "false positive": the error of rejecting a null hypothesis when it is actually true.
Type II error, also known as an "error of the second kind", a β error, or a "false negative": the error of not rejecting a null hypothesis when the alternative hypothesis is the true state of nature
Let us consider an example:
A manufacturer of handheld calculators receives very large shipments of printed circuits from a supplier. It is too costly and time-consuming to inspect all incoming circuits, so when each shipment arrives, a sample is then used to test H0: p = 0.05 versus HA: p > 0.05, where p is the true proportion of defectives in the shipment. If the null hypothesis is not rejected, the shipment is accepted, and the circuits are used in the production of the calculators. If the null hypothesis is rejected, the entire shipment is returned to the supplier due to inferior quality.
In this problem type I error is the event of rejecting a shipment based on the sample when actually the shipment contains fewer than 5% defectives.
In this problem type II error is the event of accepting a shipment based on the sample when actually the shipment contains more than 5% defectives.
2. Explain a one-tailed and two-tailed test. Use an example if needed.
In a one-tailed test we are testing for the possibility of the effect in a specific direction and completely disregard the possibility of the effect in the other direction. A one-tailed test allocates all the significance level for testing the statistical significance in the specified direction of interest. Let us consider an example in which we are comparing the mean of a sample to a given value. In a one-tailed test we test either if the mean is significantly greater than the given value or if the mean is significantly less than the given value, but not both.
In a two-tailed test, regardless of the direction of the relationship we hypothesize, we are testing for the possibility of the effect in both directions. A two-tailed test allocates half of the significance level for testing the statistical significance in one direction and the remaining half in the other direction. Let us consider an example in which we are comparing the mean of a sample to a given value. In a two-tailed test we test both if the mean is significantly greater than the given value and if the mean significantly less than the given value simultaneously.
3 Define the following terms in your own words.
• Null hypothesis
The null hypothesis (H0) is a statement about a population parameter that is assumed to be true. We will test whether the value stated in the null hypothesis is likely to be true.
• P-value
A p-value is a measure of how much evidence we have against the null hypothesis. The p-value measures consistency by calculating the probability of observing the results from your sample of data or a sample with results more extreme, assuming the null hypothesis is true.
• Critical value
Critical values are cut-off values that define regions where the test statistic is unlikely to lie. If the test statistic falls in the critical region, the hypothesis test is said to be significant.
• Statistically significant
Statistical significance, describes a decision made concerning a value stated in the null hypothesis. When the null hypothesis is rejected, we reach significance. When the null hypothesis is retained, we fail to reach significance.
4. A homeowner is getting carpet installed. The installer is ...

#### Solution Summary

The solution provides step by step method for the calculation of testing of hypothesis. Formula for the calculation and Interpretations of the results are also included.

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