Consider the time series model
where
(i) Is this a stationary time series?
(ii) What are the mean and variance of the time series?
(iii) Calculate the autocorrelation function.
(iv) Plot the correlogram.
Let's solve this step by step manually. We are given the time series: $$ y_t = 10 + 0.5y_{t-1} + \varepsilon_t, \quad \varepsilon_t \sim N(0,1) $$ --- ### (i) Stationarity A AR(1) process $y_t = c + \phi y_{t-1} + \varepsilon_t$ is stationary if $|\phi| < ___ __________ __________ _______ _____ ___.
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At a call centre, callers have to wait till an operator is ready to take their call. To monitor this process, 5 calls were recorded every hour for the 8-hour working day. The data below shows the waiting time in seconds:
| Time | Sample Number | ||||
|
9 a.m | 1 | 2 | 3 | 4 | 5 |
| 7 11 12 11 7 10 8 | 9 10 12 8 10 7 7 11 | 15 7 10 6 6 10 4 11 | 4 6 9 9 14 4 10 11 | 11 8 10 12 11 11 10 7 | |
A manufacturer of men’s jeans purchases zippers in lots of 500. The jeans manufacturer uses single-sample acceptance sampling with a sample size of 10 to determine whether to accept the lot. The manufacturer uses c = 2 as the acceptance number. Suppose 3% nonconforming zippers are acceptable to the manufacturer and 8% nonconforming zippers are not acceptable. Find
(i) Probability of accepting a lot of incoming quality 0.04.
(ii) Average outing quality (AOQ), if the rejected lots are screened and all defective zippers are replaced by non-defectives.
(iii) Average total inspection (ATI).
A manufacturer of men’s jeans purchases zippers in lots of 500. The jeans
manufacturer uses single-sample acceptance sampling with a sample size of 10 to
determine whether to accept the lot. The manufacturer uses c = 2 as the acceptance
number. Suppose 3% nonconforming zippers are acceptable to the manufacturer and
8% nonconforming zippers are not acceptable. Find
(i) Probability of accepting a lot of incoming quality 0.04.
(ii) Average outing quality (AOQ), if the rejected lots are screened and all defective
zippers are replaced by non-defectives.
(iii) Average total inspection (ATI).
) The marketing manager of a company recorded the number of mobiles sold quarterly for which are given in the following table:
| Year | Quarter | ||||
| 2018 | 48 | 41 | 60 | 65 | |
| 2019 | 58 | 52 | 68 | 74 | |
| 2020 | 60 | 56 | 75 | 78 | |
An office supply company ordered a lot of 400 printers. When the lot arrives the company inspector will randomly inspect 12 printers. If more than three printers in the sample are non-conforming, the lot will be rejected. If fewer than two printers are nonconforming, the lot will be accepted. Otherwise, a second sample of size 8 will be taken. Suppose the inspector finds two non-conforming printers in the first sample and two in the second sample. Also AQL and LTPD are 0.05 and 0.10 respectively. Let incoming quality be 4%.
(i) What is the probability of accepting the lot at the first sample?
(ii) What is the probability of accepting the lot at the second sample?
In single sampling plan, if we increase acceptance number then the OC curve will be steeper.
Differentiate between the autoregressive and moving average models of time series.