Are these statements true or false?
1. The sample mean, the sample proportion and the sample standard deviation are all unbiased estimators of the corresponding population parameters.
2. If the population is normally distributed with known variance then the sample mean may not be normally distributed for very small sample size.
Assume that thermometers are distributed with a mean of 0 degress and a standard deviation of 1. Find the probability that a randomly selected thermometer reads greater than 1.25
1. The first statement is true: The sample mean, the sample proportion, and the sample standard deviation are all unbiased estimators of the corresponding population parameters.
In order to determine the truth of this statement, we need to understand the concept of bias in statistics. Bias refers to the systematic deviation of an estimator from the true value of the population parameter it is estimating. An unbiased estimator, on the other hand, has the property that its expected value is equal to the true value of the population parameter.
The sample mean is an unbiased estimator of the population mean. This means that if we take multiple random samples from the population, calculate the means of those samples, and then average those sample means, the resulting value will be very close to the true population mean.
Similarly, the sample proportion is an unbiased estimator of the population proportion. This means that if we take multiple random samples from the population, calculate the proportions of those samples, and then average those sample proportions, the resulting value will be very close to the true population proportion.
Finally, the sample standard deviation is an unbiased estimator of the population standard deviation. This means that if we take multiple random samples from the population, calculate the standard deviations of those samples, and then average those sample standard deviations, the resulting value will be very close to the true population standard deviation.
2. The second statement is also true: If the population is normally distributed with a known variance, then the sample mean may not be normally distributed for very small sample sizes.
To understand this statement, we need to consider the Central Limit Theorem (CLT). According to the CLT, as the sample size increases, the distribution of sample means (for sufficiently large sample sizes) approaches a normal distribution, regardless of the shape of the population distribution.
However, for very small sample sizes, the distribution of sample means can deviate from the normal distribution, even if the population is normally distributed. This deviation occurs because the sample mean is sensitive to the specific values observed in the sample.
Therefore, if you have a very small sample size and the population is normally distributed with a known variance, the sample mean may not follow a normal distribution. You would need to use alternative statistical methods or consider the distribution of the sample mean in such cases.