The average temperature in households is 67.6 degrees F. The standard deviation is 4.2 degrees F. A random sample of 51 households is to be selected
What is the probability that the average of this sample will be within 1.4 STANDARD ERRORS of the population mean?
mean = 67.6
standard deviation = 4.2
sample size = 51
Standard error: sd/√n = 4.2/√51 = 0.588
Next step:
67.6 - 1.4 = 66.2
67.6 + 1.4 = 69
Use z-scores:
z = (x - mean)/sd
Therefore:
z = (66.2 - 67.6)/0.588 = -2.38
z = (69 - 67.6)/0.588 = 2.38
Finally:
Check z-table for the probability between the above two z-scores. This will be your answer.
I hope this helps.
To determine the probability that the average of the sample will be within 1.4 standard errors of the population mean, we need to calculate the standard error first.
The standard error (SE) is a measure of the variability of the sample means around the population mean. It is calculated by dividing the standard deviation by the square root of the sample size.
Given that the standard deviation is 4.2 degrees F and the sample size is 51, we can calculate the standard error using the following formula:
SE = standard deviation / square root of sample size
SE = 4.2 / √51
Calculating this, we find that the standard error is approximately 0.589 degrees F.
To determine the range within 1.4 standard errors of the population mean, we can multiply the standard error by 1.4 and then add/subtract this value from the population mean.
Range = (SE) * (1.4) = 0.589 * 1.4
Calculating this, we find that the range is approximately 0.8246 degrees F.
Now, we can calculate the probability that the average of the sample will fall within this range from the population mean.
To do this, we can use the Z-score formula. The Z-score represents how many standard deviations away a particular value is from the mean.
Z = (sample mean - population mean) / standard error
In this case, we want the probability that the sample mean falls within 1.4 standard errors from the population mean. So, we need to calculate the Z-score for 1.4 standard errors.
Z = (1.4 * standard error)
Calculating this, we find that the Z-score is approximately 0.8246.
Finally, we can use a standard normal distribution table or a statistical calculator to find the probability corresponding to this Z-score.
Using the standard normal distribution table or a calculator, we find that the probability corresponding to a Z-score of 0.8246 is approximately 0.7977 or 79.77%.
Therefore, the probability that the average of the sample will be within 1.4 standard errors of the population mean is approximately 79.77%.