Question

I have some data from a random sample of 343 students who are taking statistics at a particular college. For each student, gender is recorded along with their GPA. The mean GPA for the 182 males in the sample is 3.283. The mean GPA for the 161 females in the sample is 3.092. We wish to estimate the difference in the mean GPA’s for all males and females who take statistics at this college. The standard error of the relevant statistic is 0.043.

(a) Give notation for the parameter we are estimating.

(b) Give the best estimate of the parameter from the data. Use correct notation.

(c) The bootstrap distribution that was used to estimate the standard error is symmetric and bell-shaped. Construct a 95% confidence interval for the param- eter. Remember that the SE = 0.043.

(d) Is it plausible that the mean GPA’s for males and females taking statis- tics at this college is the same? Explain.

(e) Interpret the 95% interval in context.

(f) A standard error is a special kind of the standard deviation. What is it a standard deviation of? Also, what does the value of the standard error tell us in plain English?

Answer 1

Standard error increases when standard deviation, i.e. the variance of the population, increases. Standard error decreases when sample size increases.

Standard deviation tells you how spread out the data is. It is a measure of how far each observed value is from the mean.

If you take enough samples from a population, the means will be arranged into a distribution around the true population mean. The standard deviation of this distribution, i.e. the standard deviation of sample means, is called the standard error.

The standard error tells you how accurate the mean of any given sample from that population is likely to be compared to the true population mean. When the standard error increases, i.e. the means are more spread out, it becomes more likely that any given mean is an inaccurate representation of the true population mean.