Question

Question 1

The production manager of a camera manufacturer claims that their

latest digital camera is able to focus in just 0.0025 seconds on the

average. A random sample of the focus time in seconds for 20 cameras

is collected and the times are recorded as follows:

0.0033	0.0028	0.0021	0.0032	0.0020	0.0025	0.0026	0.0022	0.0031	0.0033
0.0022	0.0025	0.0031	0.0024	0.0021	0.0023	0.0028	0.0031	0.0028	0.0035

a) In constructing a confidence interval estimate for the true average

focus time, which probability distribution has to be used? Explain

briefly and state what assumption(s) must be made initially.

b) Construct a 90% confidence interval estimate for the true average

focus time of the camera. Keep your answers to an accuracy of six

decimal places.

c) At the 0.1 level of significance, is there any evidence to revea l that

the average focus time of the camera is more than 0.0025 seconds?

d) Use the p-value approach to test the hypothesis in part (c) at the 0.1

level of significance.

Question 2

The 802.11ax Wi-Fi standard is the latest on the market. The marketing

director of a computer equipment manufacturer believes that about 10%

of router users have switched to the new routers that support 802.11ax

Wi-Fi. A random sample of 150 router users is going to be chosen.

a) Assuming that the belief of the director is true, that is, 10% of

all router users have actually switched to the routers that support

802.11ax Wi-Fi, what is the probability that the sample proportion

of router users who have switched to these routers is:

i) less than 0.08?

ii) between 0.13 and 0.15?

b) Suppose a sample has ultimately been collected and 18 out of the

150 router users were found to have switched to the new routers.

Construct a 99% confidence interval for the true proportion of r outer

users who have switched to the new routers. Keep the answers to

four decimal places accuracy.

c) How large a random sample should actually be taken to estimate the

true proportion of router users who have switched to the new routers

to within ±0.06 with 95% confidence?

d) With reference to the information collected from the sample given

in part (b), at the 0.05 level of significance, is there any evidence to

reveal that the proportion of router users who have switched to the

routers that support the 802.11ax Wi-Fi is equal to 10%?

e) Use the p-value approach to test the hypothesis in part (d) at the

0.05 level of significance.

f) What is Type I error and Type II error in hypothesis testing? With

regard to the hypothesis testing performed in part (d), state which

type of error the result is subject to.

Question 3
In a production process, the standard weight of a product is 42 grams.

From past records, the weight of the product is a random variable

having a normal distribution with a mean of 42 grams and a standard

deviation of 0.5 grams.

a) What is the probability that a randomly selected product will weigh

less than 40.8 grams?

b) A random sample of 16 products was taken.

i) What are the mean and the standard deviation of the

corresponding sample mean?

ii) What is the probability that the sample mean weight of the 16

products will be between 41.75 and 42.15 grams?

iii) Find the weight that would be exceeded by 90% of the sample

means.

Answer 1

Q 1)

a) Because the population standard deviation is unknown, we use t distribution for one sample mean to find confidence interval.

b) Sample Mean is

Sample Standard deviation is s = 0.00047

degrees of freedom = n -2 = 18

90% CI for true mean is

( 0.002504, 0.002886 )

c) Since the CI does not inlcude 0.0025, hence there is a evidence to revea l that the average focus time of the camera is more than 0.0025 seconds

Under H0, the test statistic is

Critical value of t is 1.328

Since t calculated is greatre than t tabulated. REject H0

Hence, there is a evidence to revea l that the average focus time of the camera is more than 0.0025 seconds

d) The P-Value is .035742

Since p value is less than signiifcance level, Reject H0