(i) Curve of F-distribution
(ii) Mean Squared Error
(i) **Curve of F-Distribution**:
The F-distribution, also known as the Fisher-Snedecor distribution, is a probability distribution that arises frequently in statistics, particularly in the context of comparing variances or testing hypotheses about variances. The curve of the F-distribution is characterized by its shape, which is typically right-skewed and non-negative.
1. **Shape**:
The shape of the F-distribution is determined by two degrees of freedom parameters, \( d_1 \) and \( d_2 \). These parameters dictate the spread and shape of the distribution. As the degrees of freedom increase, the F-distribution becomes more symmetric, resembling a normal distribution.
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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 |
Complete the following table, one is done for you:
| S. No . | Test For | Name of the Test | Test Statistic | |||||||||||||
| 1 | Population mean when population variance is known and population is normal | Z-test |  Bitmap  Bitmap  Bitmap
| |||||||||||||
| 2 | Population mean when population variance is unknown and population is normal | |||||||||||||||
| 3 | Difference of two population means when samples are paired, and population of differences follows normal distribution. | |||||||||||||||
| 4 | Difference of two population means when samples are independent, and population of differences follows normal distribution. | |||||||||||||||
| 5 | Population variance when the population is normal distributed | |||||||||||||||
| 6 | Population variance when the population is not normal distributed | |||||||||||||||
Describe the various forms of the sampling distribution of ratio of two sample variances.
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
(i) Curve of F-distribution
(ii) Mean Squared Error
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.
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.