Exam 13: Introduction to Inference

The distribution of total body protein in healthy adult men is approximately Normal, with standard deviation σ= 0.1 kg. Researchers found that the mean total body protein x̄ of a random sample of 58 healthy adult men is 12.3 kg. Based on these data, what is a 95% confidence interval for μ?

A company manufactures U-100 insulin syringes designed to contain 1 milliliter (ml) of a solution containing insulin. The actual distribution of solution volumes in these syringes is Normal, with mean? and standard deviation ?= 0.05 ml. We randomly select 8 syringes and measure the volume of solution in each. The results of these 8 measurements (in ml) are as follows: 1.05 1.04 1.06 1.01 0.98 0.98 1.03 0.99 Do these data give evidence that the true population mean solution volume is not 1 m1? What is the test statistic for this hypothesis test?

A company manufactures U-100 insulin syringes designed to contain 1 milliliter (ml) of a solution containing insulin. The actual distribution of solution volumes in these syringes is Normal, with meanμ and standard deviation σ= 0.05 ml. We randomly select 8 syringes and measure the volume of solution in each, and ask if the resulting data give evidence that the true population mean solution volume is not 1 ml. Here is the software output from a one-sample z test on the data collected: Variable   N   Mean   StDev   SE   Mean   95%   CI   Z   P Volumes   8   1.0175   0.0320   0.0177   (0.9829, 1.0521)   0.99   0.322 ​ What is the appropriate alternative hypothesis for this test?

A medical researcher treats 100 subjects with high cholesterol levels with a new drug. The average decrease in cholesterol level is x̄= 80 after two months of taking the drug. Assume that the decrease in cholesterol after two months of taking the drug follows a Normal distribution, with unknown mean μ and standard deviation σ= 20. Which of the following would produce a confidence interval with a smaller margin of error than the 90% confidence interval you computed in Question 8?

The distribution of total body protein in adult men with liver cirrhosis is approximately Normal, with standard deviation σ= 0.1 kg. Researchers found that the mean total body protein x̄ of a random sample of 67 adult men with liver cirrhosis is 10.3 kg. Based on these data, what is the margin of error for a 95% confidence interval for μ?

Which of the following would be strong evidence against the null hypothesis?

What is the P-value of a test of the null hypothesis?

The scores of a certain population on the Wechsler Intelligence Scale for Children IV (WISC IV) are thought to be Normally distributed, with mean μ and standard deviation σ= 15. Bill is a child psychologist who obtains a simple random sample of 25 children from this population; each child is given the WISC IV. Bill wishes to test whether the mean for this population differs from the national average of 95, so he uses the hypotheses H₀: = 95 and H_a: μ=95, based on an SRS of size 25 from the population. Bill then calculates a μ≠95% confidence interval for μ and finds it to be 98.42 to 110.20. What should he conclude?

What is the upper 0.05 critical value of the standard Normal distribution?

A 95% confidence interval for the mean lead concentration in the urine of adult men working with lead (for smelting) is 8.22 to 11.98 micrograms per liter ( μ g/l). The numerical value of the margin of error for this confidence interval is __________μ g/l.

You measure the lifetime of a random sample of 25 rats that are exposed to 10 Sv of radiation (the equivalent of 1000 REM), for which the LD₁₀₀ is 14 days. The sample mean is x?= 13.8 days. Suppose that the lifetimes for this level of exposure follow a Normal distribution, with unknown mean ? and standard deviation ?= 0.75 day. Suppose you had measured the lifetimes of a random sample of 100 rats rather than 25. Which of the following statements is TRUE?

You measure the lifetime of a random sample of 25 rats that are exposed to 10 Sv of radiation (the equivalent of 1000 REM), for which the LD₁₀₀ is 14 days. The sample mean is x̄= 13.8 days. Suppose that the lifetimes for this level of exposure follow a Normal distribution, with unknown mean μ and standard deviation σ= 0.75 day. What is the 99% confidence interval for μ based on these data?

A company manufactures U-100 insulin syringes designed to contain 1 milliliter (ml) of a solution containing insulin. The actual distribution of solution volumes in these syringes is Normal, with meanμ and standard deviation σ= 0.05 ml. We randomly select 8 syringes and measure the volume of solution in each, and ask if the resulting data give evidence that the true population mean solution volume is not 1 ml. Here is the software output from a one-sample z test on the data collected: Variable   N   Mean   StDev   SE   Mean   95%   CI   Z   P Volumes   8   1.0175   0.0320   0.0177   (0.9829, 1.0521)   0.99   0.322 ​ Using a significance level of 0.05, what should you conclude?

A 95% confidence interval for the true mean cholesterol level of adult males based on 25 randomly selected subjects extends from 175 mg/l to 250 mg/l. What is a proper interpretation of the confidence interval?

The level of nitrogen oxides (NO_x) in the exhaust of cars of a particular model varies Normally, with standard deviation σ= 0.05 gram per mile (g/mi). A random sample of 12 cars of this particular model is taken and is found to have a mean NO_x emission of x̄= 0.298 g/mi. Government regulations call for NO_x emissions no higher than 0.3 g/mi. Do the data provide evidence that this particular model meets the government regulation? What is the test statistic for the appropriate null and alternative hypotheses?

The level of calcium in the blood of healthy young adults follows a Normal distribution, with mean μ= 10 milligrams per deciliter and standard deviation σ= 0.4. A clinic measures the blood calcium of 25 healthy pregnant young women at their first visit for prenatal care. The mean of these 25 measurements is x̄= 9.6. Is this evidence that the mean calcium level in the population from which these women come is less than 10? To answer this, test the following hypotheses: H₀: μ= 10 versus H_a: μ< 10 If the P-value of your test is less than 0.0002, what would you conclude?

A company manufactures U-100 insulin syringes designed to contain 1 milliliter (ml) of a solution containing insulin. The actual distribution of solution volumes in these syringes is Normal, with mean? and standard deviation ?= 0.05 ml. We randomly select 8 syringes and measure the volume of solution in each. The results of these 8 measurements (in ml) are as follows: 1.05 1.04 1.06 1.01 0.98 0.98 1.03 0.99 Do these data give evidence that the true population mean solution volume is not 1 m1? What is the appropriate alternative hypothesis for this test?

Is the mean age at which U.S. children learn to walk less than 15 months? A study of 40 U.S. children found a mean walking age of x̄= 13.2 months. If the population of all U.S. children has mean age μ until they begin to walk and standard deviation σ, which of the following null and alternative hypotheses should we test to answer this question?

The distribution of total body protein in healthy adult men is approximately Normal, with standard deviation σ= 0.1 kg. Researchers found that the mean total body protein x̄ of a random sample of 58 healthy adult men is 12.3 kg. Based on these data, what is a 99% confidence interval for μ?

You measure the lifetime of a random sample of 25 rats that are exposed to 10 Sv of radiation (the equivalent of 1000 REM), for which the LD₁₀₀ is 14 days. The sample mean is x̄= 13.8 days. Suppose that the lifetimes for this level of exposure follow a Normal distribution, with unknown mean μ and standard deviation σ= 0.75 day. A 95% confidence interval for is ______________ the 99% confidence interval for μ.