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STAT 350 Lecture Notes - Lecture 3: Point Estimation, Confidence Interval, Null Hypothesis

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School
George Mason University
Department
Statistics
Course
STAT 350
Professor
David Holmes

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Document Summary

Inference on proportions (one sample proportion -- confidence intervals, determining sample size and hypothesis. Mean of sampling distribution of p hat is p; that is, p hat is an unbiased estimator. For large samples, the sampling distribution of p hat is approximately normal. a sample is considered large if n p hat is greater than or equal to fifteen and (1-p hat) is greater than or equal to 15. In words: point estimate +/- (critical value) x (standard deviation of the sampling. The formula under the square root is the standard error. Z* is standard normal value from tables. Hypothesis testing (one sample) level for the test. Direction of ha depends on the research question. State null and alternative hypotheses and the value of alpha, the significance. Use statistic to make decision regarding the null hypothesis. Interpret statistic in context of the researcher"s initial question. Consider inferential procedures concerning a comparison between the rates of.

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Related textbook solutions

Introductory Statistics
OER Edition, 2013
Openstax
ISBN: 9781938168208

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Related Questions

6. As the test statistic becomes larger, the p-value

(a) gets smaller

(b) becomes larger 

(c) stays the same, since the sample size has not been changed

(d) becomes negative.

 

7. To describe the sampling distribution of the sample proportion, we use

(a) The sample proportion 

(b) The population proportion 

(c) The population mean

(d) The population mean

 

8. When small samples are used to estimate a population mean, in cases where the population standard deviation is unknown and the original distribution is normal:

(a) There always will be a large amount of sampling error.

(b) the t-distribution must be used to obtain the critical value.

(c) the resulting margin of error for a confidence interval estimate will tend to be fairly small.

(d) the z distribution must be used to obtain the critical value.

 

 

Never forget that even small effects can be statistically significant if the samples are large. During a three-year period, 8 of the 57 headed by men and 6 of the 35 headed by women failed.

(a) Find the proportions of failures for businesses headed by women and businesses headed by men. These sample proportions are quite close to each other. Give the p-value for the hypothesis that the same proportion of women's and men's businesses fail. (Use two-sided alternative). What can we conclude (use alpha as 0.05)?

The p-value was ___________ so we conclude that ____________.

 

(b) Now suppose that the same sample proportion came from a sample 30 times as large. That is 180 out of 1050 businesses headed by women and 240 out of 1710 businesses headed by men fail. Verify that the proportions of failures are exactly the same as in part (a). Repeat the test for the new data. What can we conclude?

The p-value was ___________ so we conclude that ____________.

c) It is wise to use a confidence interval to estimate the size of an effect rather than just giving a p-value. Give 95% confidence intervals for the difference between proportions of men's and women's businesses that fail for the settings of both (a) and (b).

Interval for smaller samples: ___________ to ___________.

Interval for larger samples: ___________ to ___________.

What is the effect of larger samples on the confidence interval?

 

Use the following scenario for questions 6 – 8. In 2016-17, the average SAT score of the UVA first year class was 1420 out of 1600. An advisor in the statistics department is interested in determining whether statistics majors from that class have higher SAT scores than the average. He tests H,:µ = 1420 against Hạ: µ > 1420, where u is the mean SAT score for statistics majors from the 2016-17 first year class. In a simple random sample of 30 statistics majors, he finds a sample mean SAT score of 1440 with sample standard deviation of 40.

6. What is the value of the test statistic for this significance test?

a. 0.09

b. 0.5

c. 2.73

d. 20

7. From what distribution is this test statistic?

a. N(0, 1)

b. N(1420, 7.30)

c. t distribution with 29 degrees of freedom

d. t distribution with 28 degrees of freedom

8.) The p-value for this test is 0.005. Which of the following is a proper interpretation of this p- value?

a.There is a 0.5% chance that the null hypothesis is true.

b. If the population mean SAT score really is 1420, there's a 0.5% chance that the sample a. mean SAT score from 30 statistics majors would equal 1440.

c.If the population mean SAT score really is 1420, there's a 0.5% chance that the sample mean SAT score from 30 statistics majors would be 1440 or lower.

d. If the population mean SAT score really is 1420, there's a 0.5% chance that the sample mean SAT score from 30 statistics majors would be 1440 or higher.