Describe the various forms of the sampling distribution of ratio of two sample variances.
The sampling distribution of the ratio of two sample variances is known as the F-distribution, named after Sir Ronald A. Fisher. The F-distribution is a continuous probability distribution that arises in the context of analyzing the ratio of two independent chi-square variates, each divided by their respective degrees of freedom.
The F-distribution has the following forms:
- Central F-distribution:
- This form of the F-distribution arises when the null _________ ____ _______ ____ ________.
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- This form of the F-distribution arises when the null _________ ____ _______ ____ ________.
The department of transportation has mandated that the average speed of cars on interstate
highways be no more than 70 km per hours in order. To check that the people follow it or not,
a researcher took a random sample of 186 cars and found that the average speed was 72 km
per hours with a standard deviation 0.6 km per hours.
(a) Construct the interval around the sample mean that would contain the population mean 95% of the time.
(b) If the researcher wants to test that the true mean speed on its highways is 70 km per hours or less with 95% confidence then
(i) State null and alternative hypotheses.
(ii) Name the test which is suitable in this situation and why?
(iii) Calculate the value of test statistic and critical value.
(iv)Draw the conclusion on the basis of the applied test
An engineer conducted an experiment to compare two metals: iron and copper, as bonding agents for an alloy material. Components of the alloy were bonded using the metals as bonding agents, and the pressures required to break the bonds were measured. The data for the breaking pressures are given in the following table:
| S. No. | Breaking Pressure | |
| Iron | Copper | |
| 1 | 72.7 | 73 |
| 2 | 69.6 | 67.2 |
| 3 | 83.4 | 75.3 |
| 4 | 78.9 | 61.4 |
| 5 | 75 | 74 |
| 6 | 71.6 | 69.5 |
| 7 | 85.7 | 69.8 |
| 8 | 73.5 | 73.8 |
| 9 | 70.4 | 68 |
| 10 | 84.2 | 76.1 |
(i) Curve of F-distribution
(ii) Mean Squared Error
Describe the various forms of the sampling distribution of ratio of two sample variances.
A baby-sister has 6 children under her supervision. The age of each child is as follows:
| Child | Age (in years) | |
| Sonu | 2 | |
| Lavnik | 4 | |
| Chiya | 3 | |
| Amam | 3 | |
| Avishi | 4 | |
| Ridhi | 5 | |
| Sidhi | 3 | |
A sample of 500 shops was selected in a large metropolitan area to determine various information concerning consumer behaviour. One question, among the questions, asked, was “Do you enjoy shopping for clothing?” Out of 240 males 136 answered yes. Out of 260 females, 224 answered yes. Find 95% confidence interval for the difference of the proportions for enjoys shopping for clothing.
State whether the following statements are True or False. Give reason in support of your answer:
(i) if and
is a random sample of size 5 taken from an Exponential distribution, then estimator T1 is more efficient than
.
(ii) if and
are two estimators of the parameter
such thet Var
and Var
then
is more efficient then
(iii) A 95% confidence interval is smaller than 99% confidence interval.
(iv) If the probability density function of a random variable X follows F-distribution is
then degrees of freedom of the distribution will be (2,2).
(v) A patient suffering from fever reaches to a doctor and suppose the doctor formulate the hypotheses as
The patient is a chikunguniya patient
The patient is not a chikunguniya patient
If the doctor rejects H0 when the patient is actually a chikunguniya patient, then the doctor commits type II error.