# statistics problems

1. It is estimated that 27% of all California adults are college graduates and that 31% of California adults are regular internet users. It is also estimated that 21% of California adults are both college graduates and regular internet users.

(a) Among California adults, what is the probability that a randomly chosen internet user is a college graduate? Round your answer to 2decimal places.

(b) What is the probability that a California adult is an internet user, given that he or she is a college graduate? Round your answer to 2 decimal places.

2. A soft drink company has recently received customer complaints about its one-liter-sized soft drink products. Customers have been claiming that the one-liter-sized products contain less than one liter of soft drink. The company has decided to investigate the problem. According to the company records, when there is no malfunctioning in the beverage dispensing unit, the bottles contain 1.01liters of beverage on average, with a standard deviation of 0.14liters. A sample of 70bottles has been taken to be measured from the beverage dispensing lot. The mean amount of beverage in these 70bottles was 0.995liters. Find the probability of observing a sample mean of 0.995liters or less in a sample of 70bottles, if the beverage dispensing unit functions properly.

Carry your intermediate computations to at least four decimal places. Round your answer to at least three decimal places.

3. An existing inventory for a test measuring self-esteem indicates that the scores have a standard deviation of 8 . A psychologist gave the self-esteem test to a random sample of 70individuals, and their mean score was 68 . Construct a 95%confidence interval for the true mean of all test scores. Then complete the table below.

Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place.

This goes with question 3

What is the lower limit of the 95% confidence interval?

What is the upper limit of the 95% confidence interval?

4. Use the calculator provided to solve the following problems.

- Consider a
*t*distribution with 15degrees of freedom. Compute P(t > 1.33). Round your answer to at least three decimal places.

Consider a *t* distribution with 18degrees of freedom. Find the value of c such that

P(-c<t<c) = 0.90. Round your answer to at least three decimal places.

5. The records of a casualty insurance company show that, in the past, its clients have had a mean of 1.7auto accidents per day with a variance of 0.0036. The actuaries of the company claim that the variance of the number of accidents per day is no longer equal to 0.0036. Suppose that we want to carry out a hypothesis test to see if there is support for the actuaries’ claim. State the null hypothesis H0and the alternative hypothesis H1 that we would use for this test.

6. A leasing firm claims that the mean number of miles driven annually, **u**, in its leased cars is less than 12740miles. A random sample of 100cars leased from this firm had a mean of 12419annual miles driven. It is known that the population standard deviation of the number of miles driven in cars from this firm is 1540miles. Is there support for the firm’s claim at the .05level of significance?

Perform a one-tailed test. Then fill in the table below.

The null hypothesis. H0

The alternative hypothesis H1

The type of statistic: Z, t, F, or Chi square

The value of the test statistic (round to at least three decimal places.

The p-value (Round to at least three decimal places

Can we support the leasing firms claim that the mean number of miles driven annually is less than 12740 miles? Yes or No

7. Random and independent samples of 85recent prime time airings from each of two major networks have been considered. The first network aired a mean of 110.6 commercials during prime time, with a standard deviation of 4.4commercials. The second network aired a mean of 109.4commercials, with a standard deviation of 4.5commercials. As the sample sizes are quite large, the population standard deviations can be estimated using the sample standard deviations. Construct a 90%confidence interval for **u**1 – **u**2, the difference between the mean number of commercials **u**1aired during prime time by the first network and the mean number of commercials **u**2aired during prime time by the second network. Then complete the table below.

Carry your intermediate computations to at least three decimal places. Round your answers to at least two decimal places.

8. A random selection of volunteers at a research institute have been exposed to a typical cold virus. After they started to have cold symptoms, 15of them were given multivitamin tablets daily which contain 3grams of vitamin C and various other vitamins and minerals. The remaining 15volunteers were given placebo tablets. For each individual, the length of time taken to recover from cold is recorded. At the end of the experiment following data are obtained:

**Treated with multivitamin – 4.4; 4.6; 5.5; 4.9; 5.1; 7.7; 6.5; 7.7; 4.5; 5.8; 7.9; 7.2; 3.3; 4.6; 4.7**

**Treated with placebo – 1.8 3.6 5.6 2.4 3.7 4.8 4.0 5.0 3.8 4.9 4.7 4.9 1.7 6.3 6.4**

It is known that the population standard deviation of recovery time from cold is 1.8days when treated with multivitamin, and the population standard deviation of recovery time from cold is 1.5days when treated with placebo tablets. It is also known that both populations are approximately normally distributed. The researchers claim that the mean recovery time, **u****1**, of the patients treated with multivitamin is not equal to the mean recovery time **u**2, of the patients who are treated with placebo tablets. At the .05level of significance, is there enough evidence to support this claim? Perform a two-tailed test. Then fill in the table below.

The null hypothesis. H0

The alternative hypothesis H1

The type of statistic: Z, t, F, or Chi square

The value of the test statistic (round to at least three decimal places.

The p-value (Round to at least three decimal places

Can we support the researchers claim that the mean recovery time when treated with multivitamin is not equal to the mean recovery time when treated with placebo? Yes or No

9. Use the calculator provided to solve the following problems:

- Consider an
*F*distribution with 3numerator degrees of freedom and 48denominator degrees of freedom. Compute P(F<0.90). Round your answer to at least three decimal places.

Consider an *F* distribution with 28numerator degrees of freedom and 24denominator degrees of freedom. Find csuch that P(>c) = 0.1. Round your answer to at least two decimal places.

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