Math Problem Solver

An urn contains red and white marbles in an unknown proportion. A random sample of

60 marbles selected with replacement from the urn showed that 70% were red. Find:

(i) 95% and 99% confidence limits for the actual proportion of red marbles in the urn.

(ii) size of the sample of marbles in order to be 95% and 99% confident that the true

sample proportions do not differ more than 5%.

2. Two samples of sizes 16 and 10, respectively, are drawn at random from two normal

populations. If their variances are found to be 24 and 18 respectively, find (a) 98% and

(b) 90% confidence limits for the ratio of the variances.

3. The standard deviation of the breaking strengths of 10 cables tested by a company was

815 kilograms. Find 95% and 99% confidence limits for the standard deviation of all

cables produced by the company.

4. The standard deviation of the lifetimes of a sample of 200 electric light bulbs was

computed to be 100 hours. How large a sample must we take in order to be 99%

confident that the true population standard deviation will not differ from the sample

standard deviation by more than (a) 5% and (b) 10%?

5. In the past, the standard deviation of weights of certain 1135-gram packages filled by a

machine was 7.1 grams. A random sample of 20 packages showed a standard deviation

of 9.1 grams. Is the apparent increase in variability significant at the (a) 0.05 and (b) 0.01

level of significance?

6. A pair of dice is tossed 100 times, and it is observed that “sevens” appear 23 times. Test

the hypothesis that the dice are fair, i.e., not loaded, using (a) a two-tailed test and (b) a

one-tailed test, both with a significance level of 0.05. Discuss your reasons, if any, for

preferring one of these tests over the other.

7. A manufacturer claimed that at least 95% of the equipment, which he supplied to a

factory, confirmed to specifications.

An examination of a sample of 200 pieces of

equipment revealed that 18 were faulty. Test his claim at a significance level of (a) 1%

(b) 5%.

8. The annual temperature of a city is obtained by finding the mean temperatures on the 15th

day of each month. The standard deviation of the annual temperatures of the city over a

period of 100 years was 8.9 °C. During the last 15 years, the standard deviation of

annual temperature was computed as 5.6 °C. Test the hypothesis that the temperatures in

the city have become less variable than in the past, using the significance level of 5%.

9. A usual complaint of interactive system users is the large variance of the response time.

While contemplating the purchase of a new interactive computer system, we measure 30

random samples of response time, and the sample variance is computed to be 25 ms2.

Assuming the response times are approximately normally distributed, find a 95%

confidence interval for the population variance.