If f is a function from set X to a set Y then is it possible ,
? where
and
If and
be two independent gamma distributions and
and
then find the distribution of V.
Explain the procedure of assigning probability in the discrete world of probability theory.
See Answer →In an election there are two candidates. Being a statistician, you are interested in predicting the result of the election. So, you plan to conduct a survey. Using the learning skill of this course answer the following question. How many people should be surveyed to be at least 95% sure that the estimate is within 0.05 of the true value?
See Answer →Solve the problem in part (a) simply by replacing 10 minutes by 5 minutes all other things are the same.
See Answer →a) Suppose two friends Anjali and Prabhat trying to meet for a date to have lunch say between 1 pm to 2 pm. Suppose they follow the following rules for this meeting:
Each of them will arrive either on time or 10 minutes late or 20 minutes late or 30 minutes late or 40 minutes late or 50 minutes late or 1 hour late. All these arrival times are equally likely for both of them.
Whoever of them reaches first will wait for the other to meet only for 10 minutes. If within 10 minutes the other does not reach, he/she leaves the place and they will not meet.
Find the probability of their meeting.
Differentiate between the autoregressive and moving average models of time series.
See Answer →The marketing manager of a company recorded the number of mobiles sold quarterly for which are given in the following table:
I’ve transcribed the table from your image exactly as it appears.
Quarterly Data Table
| Year / Quarter | $Q_1$ | $Q_2$ | $Q_3$ | $Q_4$ |
|---|---|---|---|---|
| 2018 | 48 | 41 | 60 | 65 |
| 2019 | 58 | 52 | 68 | 74 |
| 2020 | 60 | 56 | 75 | 78 |
(i) Find the quarterly seasonal indexes for the mobile sold using the ratio to trend method.
(ii) Do seasonal forces significantly influence the sale of mobile? Comment.
(iii) Also find the deseasonalised values.
See Answer →The marketing manager of a company recorded the number of mobiles sold quarterly for which are given in the following table:
I’ve transcribed the table from your image exactly as it appears.
Quarterly Data Table
| Year / Quarter | $Q_1$ | $Q_2$ | $Q_3$ | $Q_4$ |
|---|---|---|---|---|
| 2018 | 48 | 41 | 60 | 65 |
| 2019 | 58 | 52 | 68 | 74 |
| 2020 | 60 | 56 | 75 | 78 |
(i) Find the quarterly seasonal indexes for the mobile sold using the ratio to trend method.
(ii) Do seasonal forces significantly influence the sale of mobile? Comment.
(iii) Also find the deseasonalised values.
See Answer →Consider the time series model
y₁=5+0.6y1-0.32 +8
where &, ~N[0,1]
(i) State whether the process is stationary, giving reasons.
(ii) Find the mean of the process.
(iii) Obtain the variance of the process.
(iv) Derive the autocorrelation function up to lag 2 and sketch the correlogram.
See Answer →In a manufacturing process, 4 items are inspected every hour for 10 consecutive hours.
The measured quality characteristic (in mm) is given below:
The measured quality characteristic (in mm) is given below.
| Hour | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 50 | 52 | 51 | 49 |
| 2 | 51 | 50 | 53 | 52 |
| 3 | 49 | 50 | 48 | 51 |
| 4 | 52 | 54 | 53 | 51 |
| 5 | 50 | 49 | 51 | 50 |
| 6 | 53 | 52 | 54 | 55 |
|---|---|---|---|---|
| 7 | 48 | 49 | 50 | 47 |
| 8 | 51 | 52 | 50 | 51 |
| 9 | 54 | 55 | 53 | 54 |
| 10 | 50 | 51 | 49 | 50 |
(i) Construct conreol chart for variability and mean and comment on the state of statistical control. If the process is out of control, obtain revised control limits.
(ii) Given specification limits 50 ± 3, compute the process capability index (Cpk) and interpret the result.
See Answer →In a manufacturing process, 4 items are inspected every hour for 10 consecutive hours.
The measured quality characteristic (in mm) is given below:
The measured quality characteristic (in mm) is given below.
| Hour | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 50 | 52 | 51 | 49 |
| 2 | 51 | 50 | 53 | 52 |
| 3 | 49 | 50 | 48 | 51 |
| 4 | 52 | 54 | 53 | 51 |
| 5 | 50 | 49 | 51 | 50 |
See Answer →
A system consists of six independent components arranged as follows:
Two components with reliabilities 0.9 and 0.8 connected in series.
This series combination is connected in parallel with a component of reliability 0.7.
The resulting subsystem is connected in series with two components of reliabilities 0.85 and 0.95.
Draw the reliability block diagram and calculate the overall reliability of the system.
See Answer →A factory purchases bolts in lots of 800. Acceptance is decided using a single-sampling plan with sample size n = 20 and acceptance number c = 3. Assume that 2% defective items are considered acceptable quality and 7% defective items are considered unacceptable quality.
Find:
(i) The probability of accepting a lot when the incoming quality level is 5% defective.
(ii) The Average Outgoing Quality (AOQ), assuming rejected lots are completely screened and defectives are replaced.
(iii) The Average Total Inspection (ATI).
See Answer →A factory purchases bolts in lots of 800. Acceptance is decided using a single-sampling plan with sample size n = 20 and acceptance number c = 3. Assume that 2% defective items are considered acceptable quality and 7% defective items are considered unacceptable quality.
Find:
(i) The probability of accepting a lot when the incoming quality level is 5% defective.
(ii) The Average Outgoing Quality (AOQ), assuming rejected lots are completely screened and defectives are replaced.
(iii) The Average Total Inspection (ATI).
See Answer →State whether the following statements are True or False. Give reason in support of your answer:
(i) The c-chart is suitable for moniterting to proportion of defective.
(ii) In single sampling plan, if we increase acceptance number then the OC curve will be steeper.
(iii) Autocorrelation measures the relationship between yt and yt-k after removing the effect of intermediate lags.
(iv) If the effect of summer and winter is not constant on the sale of AC then we use the additive model of the time series.
(v) In a series system, improving the reliability of the weakest component gives the maximum improvement in system reliability.
See Answer →Describe the following:
(i) t-distribution
(ii) General procedure of Testing of hypothesis
See Answer →
Let X1, X2, ..., Xbe a random sample from the Exponential distribution with pdf f(x; 0) = 0e-ex, x > 0, 0 > 0
(i) Obtain the maximum likelihood estimator (MLE) of 0.
(ii) Hence, find the maximum likelihood estimate of for the observed sample: 2.1, 1.6, 3.4, 0.9, 2.5
See Answer →A researcher wants to compare the average test scores of students taught using Method A and Method B. A random sample of students was selected from each group, and their scores are given below:
| Student | Method A | Method B |
|---|---|---|
| 1 | 68 | 72 |
| 2 | 74 | 75 |
| 3 | 71 | 78 |
| 4 | 69 | 70 |
| 5 | 73 | 76 |
| 6 | 70 | 74 |
| 7 | 72 | 77 |
| 8 | 75 | 79 |
Assuming that the populations are normally distributed with equal variances. For testing whether there is a significant difference between the mean scores of the two teaching methods at the 5% level of significance
(i) State the null and alternative hypotheses.
(ii) Name the appropriate t-test and justify your choice.
(iii) Compute the test statistic and critical value.
(iv) State your conclusion.
See Answer →A survey was conducted to study the preference for online learning among students from urban and rural areas. Out of 300 urban students, 210 preferred online learning, while out of 250 rural students, 150 preferred online learning. Construct a 95% confidence interval for the difference between the population proportions of students preferring online learning in urban and rural areas.
See Answer →