In 2013, Gallup conducted a poll and found a 95% confidence interval of the proportion of Americans who believe it is the government’s responsibility for health care. Give a statistical interpretation of this interval.

In 2013, Gallup conducted a poll and found a 95% confidence interval of the proportion of Americans who believe it is the government’s responsibility for health care.  Give a statistical interpretation of this interval.

Suppose you compute a confidence interval with a sample size of 100. What will happen to the confidence interval if the sample size decreases to 80?

Suppose you compute a confidence interval with a sample size of 100.  What will happen to the confidence interval if the sample size decreases to 80?

Maintaining your balance may get harder as you grow older. A study was conducted to see how steady the elderly is on their feet. They had the subjects stand on a force platform and have them react to a noise. The force platform then measured how much they swayed forward and backward, and the data is in the table below.

Maintaining your balance may get harder as you grow older.  A study was conducted to see how steady the elderly is on their feet.  They had the subjects stand on a force platform and have them react to a noise.  The force platform then measured how much they swayed forward and backward, and the data is in the table below.

Table: Forward/backward Sway (in mm) of Elderly Subjects

Set up and run a test of hypothesis to determine if the elderly sway more than the mean forward sway of younger people, which is 18.125 mm?  Use a level of significance of 5%. Specifically, do the following:

(i) Write down the null and alternative hypotheses.

(ii) Verify that the assumptions necessary to apply the test are satisfied.

(iii) What is the test statistic and what is the p-value?

(iv) State your conclusion.

The economic dynamism, which is the index of productive growth in dollars for countries that are designated by the World Bank as middle-income are in the table below. Countries that are considered high-income have a mean economic dynamism of 60.29.

Table: Economic Dynamism ($) of Middle Income Countries

(a) Set up and run a test of hypotheses to determine if the data show that the mean economic dynamism of middle-income countries is less than the mean for high-income countries? Use a level of significance of 5%. Specifically, do the following:

(i) Write down the null and alternative hypotheses.

(ii) Verify that the assumptions necessary to apply the test are satisfied.

(iii) What is the test statistic and what is the p-value?

(iv) State your conclusion.

(b) Using the data in the table which represents the index of productive growth in dollars for countries that are designated by the World Bank as middle-income, construct a 95% confidence interval for the true mean economic dynamism of middle-income countries. Make sure to check the assumptions necessary to construct the interval.  Interpret the interval.

Write down the null and alternative hypotheses.

In 2008, across the United States, 1 child in 88 is diagnosed with Autism Spectrum Disorder (ASD).

But in the state of Arizona alone, a sample of 32,601 children was taken and 507 were diagnosed with ASD.

(a) Set up and run a test of hypothesis to determine if there sufficient data to show that the incident of ASD is more in Arizona than nationally?  Use a level of significance of 1%. Specifically, do the following:

(i) Write down the null and alternative hypotheses.

(ii) Verify that the assumptions necessary to apply the test are satisfied.

(iii) What is the test statistic and what is the p-value?

(iv) State your conclusion(b) Using the sample data from Arizona, construct a 99% confidence interval for the true population proportion in Arizona of children diagnosed with ASD. Make sure to check the assumptions necessary to construct the interval.Interpret the interval.

According to the February 2008 Federal Trade Commission report on consumer fraud and identity theft, 23% of all complaints in 2007 were for identity theft. In that year, in the state of ALASKA alone, there were 321 complaints of identity theft out of 1,432 consumer complaints.

(a) What is the random variable?

(b) What is the population parameter?

(c) Set up and run a test of hypothesis to determine if the data from ALASKA provides enough evidence to show that ALASKA had a lower proportion of identity theft than 23%. Use a level of significance of 5%. Specifically, do the following:

(i) Write down the null and alternative hypotheses.

(ii) State the type I and type II errors in this case and state a consequences of each type of error in the context of this problem.

(iii) Verify that the assumptions necessary to apply the test are satisfied.

(iv) What is the test statistic and what is the p-value?

(v) State your conclusion.

The mean cholesterol levels of women age 45-59 in Kenya, Africa is 5.1 mmol/l and the standard deviation is 1.0 mmol/l. Assume that cholesterol levels are normally distributed.

(a) What is the random variable?

(b) Find the probability that a woman age 45-59 in Kenya has a cholesterol level above 6.2 mmol/l (considered a high level).

(c) Suppose doctors decide to test the woman’s cholesterol level again and average the two values.  Find the probability that this woman’s mean cholesterol level for the two tests is above 6.2 mmol/l.

(d) Suppose doctors being very conservative decide to test the woman’s cholesterol level a third time and average the three values.  Find the probability that this woman’s mean cholesterol level for the three tests is above 6.2 mmol/l.

(e) If the sample mean cholesterol level for this woman after three tests is above 6.2 mmol/l, what could you conclude?

A random variable is normally distributed. It has a mean of 245 and a standard deviation of 21.

A random variable is normally distributed.  It has a mean of 245 and a standard deviation of 21. 

(a) Suppose we take a random sample of size 10.

(i) What is the shape of the distribution of the sample means?  Why?

(ii) What is the mean of the sample means, and what is standard deviation of the sample means?

(iii) Find the probability that a sample mean is more than 241.

(b) Suppose we take a random sample of size 35.

(i) What is the shape of the distribution of the sample means?  Why?

(ii) What is the mean of the sample means, and what is standard deviation of the sample means?

(iii) Find the probability that a sample mean is more than 241.

(c) Compare your answers in (a)(iii) and (b)(iii).  Why is one smaller than the other?

As it turns out, the actual data for annual rainfalls for Sydney, Australia are given in the table below. Based on these data, was our assumption of normality for the previous problem a correct one? Why or why not?

Table: Annual Rainfall in Sydney, Australia

The mean yearly rainfall in Sydney, Australia, is 136.9 mm and the standard deviation is 69.4mm. Assume rainfall is normally distributed.

The mean yearly rainfall in Sydney, Australia, is 136.9 mm and the standard deviation is 69.4mm. Assume rainfall is normally distributed.

(a) What is the random variable?

(b) Find the probability that the yearly rainfall is less than 100 mm.

(c) Find the probability that the yearly rainfall is more than 240 mm.

(d) Find the probability that the yearly rainfall is between 140 and 250 mm.

(e) If a year has a rainfall less than 100mm, does that mean it is an unusually dry year?  Why or why not?

(f) What rainfall amount are 90% of all yearly rainfalls more than?