question archive A city transportation company (CTC) conducted a small sample study to estimate the utilization of bus stops in the city
A city transportation company (CTC) conducted a small sample study to estimate the utilization of bus stops in the city
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A city transportation company (CTC) conducted a small sample study to estimate the utilization of bus stops in the city. Three out of 20 city areas were sampled at random and then a few bus stops within these areas. The numbers of people using the bus stop over a specified hour in a specified weekday (weekend excluded) are given in the following table:
| Sampled Area | Number of bus stops | Number of buys stops sampled | Sample average | Sample variance |
| 1 | 45 | 9 | 82 | 30 |
| 2 | 36 | 7 | 80 | 20 |
| 3 | 20 | 4 | 56 | 30 |
a. Explain what type of sample design is used here.
b. Estimate the average number of people using a bus stop in one hour for the weekday per bus stop in the city. Is this estimator unbiased?
c. Place a bound on the error of the estimation.
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A) This type of study design is referred to as random cluster sampling. The cities represent existing clusters that are then randomly selected by the researcher.
B) To obtain an estimate of the number of people who used a bus stop in each city area, per hour, examine the column of the table labeled Sample Average. The sample average can refer to a sample mean. The sample mean is an ideal point estimate for statistical analysis and is your best guess of the answer to the question of how many people used the bus stop per hour in each listed city area.
In this instance, data from three city areas is available, take the average of these three city areas to obtain an overall mean value to represent all city areas: 82 + 80 + 56 = 218/3 = 72.7. As random cluster sampling was used to obtain the sample data, the results are not biased in this regard. However, some bias is always present in the process of data collection. For example, city events vary from day to day. Traffic will vary from day to day, week to week, and month to month, depending on the weather of the season. This means that transit usage will vary as well, introducing potential bias into the sample.
C) To place boundaries on our estimate, we will calculate a 95% confidence interval for our point estimate. In order to do so, we will first obtain a value for the pooled standard error:
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