A researcher wants to estimate the average time (in minutes) spent in a car daily. This will be done by randomly surveying 300 individuals and calculating the sample mean. Find the probability that the estimate will be in error by less than 5 minutes. Assume that the population standard deviation is 45 minutes.
To find the probability that the estimate will be in error by less than 5 minutes, we need to calculate the standard error of the sample mean.
The standard error (SE) of the sample mean is calculated using the formula:
SE = population standard deviation / √(sample size)
In this case, the population standard deviation is given as 45 minutes, and the sample size is 300.
SE = 45 / √300
Next, we can calculate the margin of error (ME), which is the maximum amount by which the estimate may differ from the actual population mean:
ME = Z * SE
The value of Z is found using the standard normal distribution table. We want the estimate to be in error by less than 5 minutes, which means we are interested in the probability below a specific Z value.
For example, if we choose 5 minutes as the maximum error, we need to find the Z value that corresponds to a cumulative probability of 1 - (probability of being below 5 minutes). This probability is equal to 1 - (probability of being above 5 minutes). We can find this value in the standard normal distribution table.
Once we have the Z value, we can calculate the margin of error:
ME = Z * SE
Finally, we can find the probability that the estimate will be in error by less than 5 minutes by using the Z value and the standard error:
Probability = 2 * (1 - cumulative probability of Z * SE)
This is a two-sided test because we are interested in the probability of the estimate being within a range of +5 minutes or -5 minutes.
Note: The specific Z value and probability may depend on the margin of error chosen.