STATISTICS CHAPTER 8

Question

According to a study conducted,

the proportion of americans who were afraid to fly in 2012 was 0.10. A random sample of 150 americans resulted in 13 indicating that they were afraid. Explain why this is not necessarily evidence that the proportion of americans who are afraid to fly has decreased. Use sampling distribution of the sample proportion of 200 sample size of 150 to justify your answer.

Answer 1

When conducting a study and taking a random sample of the population, or even multiple random samples, the mean of the different random samples will not be the same, nor will the random sample mean be necessarily equal to the mean of the population. So, even if the mean found in the new sample (in this case, ~0.0867) is different from the previous sample mean, that does not prove that the population mean changed at all.

In order to solve this issue, we ought to do an hypothesis test, where we are trying to determine if the difference is within expected parameters, or if the deviation is too high. For this, we need the sample standard deviation.

Once we have the sample standard deviation, s, we can do a t-test (the sample size is greater than 30, so the degrees of freedom (DF) is infinity) and we can apply the formula:

t = (0.0867 - 0.1) / (s/√150)

If for instance, s = 1, then

t = -0.1629

For a t-test and DF = ∞, there is a 95% chance for the t-values to be between 1.96 and -1.96. Since

-1.96 < -0.1629 < 1.96, the default hypothesis is kept and we are within expected parameters.

Another way would be to define a confidence interval based on the standard deviation and the mean of the sample, and see whether the previous sample mean falls within that confidence interval!