Homework 6 Note: You should be able the following problems, before Tuesday: 1,2,
ID: 3152099 • Letter: H
Question
Homework 6 Note: You should be able the following problems, before Tuesday: 1,2,5,6 Due: Thursday, April 7 1. Explain in one or two sentences the following: State the two conditions in which you can use the normal distribution to construct the confidence interval? (when do you use the z-table). State the two conditions in which you can use the t- distribution to construct the confidence interval? (when do you use the t-table a) b) Explain in terms of what you know (or don't know) about the distribution of the population, population variance, sample variance and sample size? 2. A bottling company wants to do a research on the accuracy of its bottle filler. The research department at the company took a sample of 16 bottles and recorded the volume of each bottle. This information produces a mean volume of 8.1 oz for this sample. It is known that the standard deviation of the volume of all bottles is 0.03 oz and the population of volume is normal. (a) What is the distribution? Explain. (b) What is the point estimate of the mean price of all such textbooks? (c) What is the 90% margin of error. (d) Construct and interpret the 90% confidence interval for the mean volume of all bottles. 3. A researcher wants to estimate the mean of 1Q of high school students in a state. It is known that the population standard deviation of the high school students in the state is 15. How large a sample should be selected so that the estimate with a 99% confidence level is within 6 of the population mean? 4. According to an estimate, 18% of junk mail is thrown out without being opened. Suppose that this is based on a random sample of 576 pieces of junk mail. a. What is the point estimate of the corresponding population proportion? (4 pts) b. What is the 99% margin of error? (4 pts) c, construct and interpret the 99% confidence interval for the proportion of all pieces of junk mail that is thrown out without being opened? (4 pts) i. The value of t for sample size equal 17 and.025 area in the right tail of a t distribution curve. i. The value of t for sample size equal 17 and.025 area in the left tail of a t distribution curve ili. The value of t for sample size equal 13 and.05 area in the right tail of a t distribution curve. iv. The value of t for sample size equal 13 and .05 area in the left tail of a t distribution curve. 5. Sketch and find the following from the t-table:Explanation / Answer
1. a) While performing a hypothesis test for of a single population mean, the conditions under which we use the normal distribution to construct the confidence interval are:
1. The sample should be chosen using the method of Simple Random Sampling.
2. The population that is being tested is either normally distributed or the sample size is larger than 30 or both.
3. The population variance is known.
b) While performing a hypothesis test for of a single population mean, the two conditions under which we use the t-distribution to construct the confidence interval are:
1. The sample should be chosen using the method of Simple Random Sampling.
2. The population that is being tested is either normally distributed or the sample size is larger than 30 or both.
3. The population variance is not known.
2. a) Since the population is known to be normal, and the standard deviation of ALL bottles (i.e. the population of bottles) is known, we should use the normal distribution as the sampling distribution of the sample mean.
b) Since the sample mean is an unbiased estimator of population mean, the point estimate of the mean volume of all bottles is 8.1oz.
Note: There appears to be a typographical error in this question.
c) Margin of error = Critical Value * Standard Error of the Statistic
Standard error of the mean volume of bottles = s/n
Since the sample standard deviation is not provided, we can use the population standard deviation instead.
Therefore, standard error = 0.03/16 = 0.0075
Now, as a general case, we assume a two-tailed test for the population mean. The 90% critical value for the normal two-sided test = 1.64.
Therefore, the 90% margin of error = 0.0075*1.64 = 0.0123
d) Therefore, the 90% confidence interval for the mean volume of bottles is (0.03 – 0.0123, 0.03 + 0.0123)
= (0.0177, 0.0423)
This means that if multiple samples are drawn from the population and the confidence interval in each case is determined, 90% of these intervals will contain the true value of the population mean.
There are actually 5 separate questions here, with their individual sub-parts. Only the first 2 questions have been answered.
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