A sample of 30 latest returns on UTI stock reveals a mean return of $4 with a sample standard deviation of $0.13. The estimated standard error of the sample mean is:
$0.02
Calculating Estimated Standard Error of Sample Mean
The question asks for the estimated standard error of the sample mean for UTI stock returns, given the sample size and sample standard deviation.
The standard error of the sample mean ($\text{SE}_{\bar{x}}$) is a measure of how much the sample mean is likely to vary from the population mean. When the population standard deviation ($\sigma$) is unknown, we estimate the standard error using the sample standard deviation ($s$). This is often the case in real-world scenarios, like analyzing stock returns.
Key Information Provided
- Sample size ($n$): 30 latest returns
- Sample mean return: $4
- Sample standard deviation ($s$): $0.13
We need to calculate the estimated standard error of the sample mean.
Formula for Estimated Standard Error
The formula to estimate the standard error of the sample mean when the population standard deviation is unknown is:
$$\text{SE}_{\bar{x}} = \frac{s}{\sqrt{n}}$$
Where:
- $\text{SE}_{\bar{x}}$ is the estimated standard error of the sample mean.
- $s$ is the sample standard deviation.
- $n$ is the sample size.
Step-by-Step Calculation
Using the provided values, we can substitute them into the formula:
$$s = 0.13$$
$$n = 30$$
Now, plug these values into the formula:
$$\text{SE}_{\bar{x}} = \frac{0.13}{\sqrt{30}}$$
First, calculate the square root of the sample size:
$$\sqrt{30} \approx 5.4772$$
Now, divide the sample standard deviation by the square root of the sample size:
$$\text{SE}_{\bar{x}} = \frac{0.13}{5.4772} \approx 0.02373$$
Rounding the result to two decimal places (as the options suggest):
$$\text{SE}_{\bar{x}} \approx 0.02$$
The estimated standard error of the sample mean for the UTI stock returns is approximately $0.02.
Understanding the Result
An estimated standard error of $0.02 indicates the typical variability or precision of the sample mean ($4) as an estimate of the true average return of UTI stock. A smaller standard error suggests that the sample mean is a more precise estimate of the population mean.
Summary of Calculation Steps
| Step | Description | Value / Formula |
|---|---|---|
| 1 | Identify sample standard deviation ($s$) | $0.13 |
| 2 | Identify sample size ($n$) | 30 |
| 3 | Calculate square root of sample size ($\sqrt{n}$) | $\sqrt{30} \approx 5.4772$ |
| 4 | Apply the estimated standard error formula | $\text{SE}_{\bar{x}} = s / \sqrt{n}$ |
| 5 | Calculate the estimated standard error | $\text{SE}_{\bar{x}} = 0.13 / 5.4772 \approx 0.02373$ |
| 6 | Round to appropriate decimals | $0.02 |
This calculation shows that based on the sample data, the estimated standard error is $0.02.
Revision Table: Standard Error Concepts
| Term | Symbol | Definition | Used For |
|---|---|---|---|
| Population Standard Deviation | $\sigma$ | Measure of dispersion for the entire population | Calculating standard error when population data is known |
| Sample Standard Deviation | $s$ | Measure of dispersion for a sample | Estimating population standard deviation when population data is unknown |
| Standard Error of the Mean | $\text{SE}_{\bar{x}}$ or $\sigma_{\bar{x}}$ | Standard deviation of the sample mean's distribution | Measuring the precision of the sample mean as an estimate of the population mean |
| Estimated Standard Error | $\text{SE}_{\bar{x}}$ | Standard error calculated using sample standard deviation ($s$) instead of population standard deviation ($\sigma$) | Used when population standard deviation is unknown |
Additional Information: Standard Error vs. Standard Deviation
It's important not to confuse standard deviation and standard error. Both are measures of variability, but they measure different things:
- Standard Deviation ($s$ or $\sigma$): Measures the spread or dispersion of individual data points around the mean within a single dataset (either a sample or a population). For example, the sample standard deviation of $0.13 tells us about the variability of the individual UTI stock returns in the sample.
- Standard Error ($\text{SE}_{\bar{x}}$): Measures the variability of the sample mean itself from sample to sample. It tells us how much the sample mean is expected to vary if we were to take multiple samples of the same size from the same population. The standard error is smaller than the standard deviation because the mean of a sample is less variable than individual data points.
The standard error decreases as the sample size increases. This is because larger samples tend to provide a more accurate estimate of the population mean, thus reducing the variability of the sample mean across different samples.
Important Questions from Sampling Theorems
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In a cluster sampling wherein the units within same cluster are highly correlated, suppose \(S_w^2\) represents the variance within the clusters and \(S_b^2\) between clusters, then which option is correct?
