Question

a. In a study of monthly salary distribution of residents in Paris conducted in year 2015, it was found that the salaries had an average of €2200 (EURO) and a standard deviation of €550. Assume that the salaries were normally distributed.

(i): Consider sampling with sample size 64 on the above population. Compute the mean of the sampling distribution of the mean (X).

(ii): Compute the standard deviation of the sampling distribution of the mean in Question 1 above.

(iii): A random sample of 64 salaries (sample 1) was selected from the above population. What is the probability that the average of the selected salaries is above €2330?

(iv): Would the calculation you performed in Question 3 still be valid if the monthly salaries were NOT normally distributed? Why?

(b) In another study conducted in the same year (2015), the average monthly salary of residents in Bordeaux was found to be about €2353. And the standard deviation of the monthly salaries was €420. A random sample of 81 salaries (sample 2) was selected from this population.

Set 1 = Paris (2015); 2 = Bordeaux (2015)

(i): Compute the mean of X1 − X2.

(ii): Compute the standard deviation of X1 − X2.

(iii): What is the probability that the average of the salaries in the sample 1 is less than the average of the salaries in sample 2?

(c) In 2017, a study on the salary distribution of Paris residents was conducted. Assume that the salaries were normally distributed. A random sample of 10 salaries was selected and the data are listed below: 3200 3500 3000 2100 2950 2050 2440 3100 3500 2500

(i): Assume that the standard deviation of the salaries was still the same as in 2015. Estimate the average salary (in 2017) with 95% confidence. Question 9: The assumption made in Question 8 was certainly unrealistic. Estimate the average salary (in 2017) with 95% confidence again assuming that the standard deviation had changed from 2015.

(ii): Estimate the variance of monthly salaries of Paris residents (in 2015) based on the sample provided above at a 95% confidence level.

(iii): How would you interpret the result in Question 10 above?

(d) A similar study was conducted on salary distribution of Paris residents in 2019. The research team aimed to estimate the average salary. They chose the 98% confidence and assumed that the population standard deviation was the same as in 2015. Assume again that those salaries were normally distributed.

(i) If they would like the (margin of) error to be no more than €60, how large a sample would they need to select?

Answer 1

a) Let X denote the monthly salary of residents of Paris

X follows normal distribution with mean 2200 and standard deviation 550

i) initially to find this we need to find the distribution of , so as X follows normal distribution with mean 2200 and standard deviation of 550 that is variance is equal to 302500.

follows normal distribution with mean 64*2200 and variance 63*302500.

So using the change of scale property on , mean of X follows

follows normal distribtion with mean (64*2200)/64 and variance (64*302500)/64*64.

Basically

ii) To find the standard deviation of mean of x in case 1

As i have computed the variance in above case, the standard deviation is just the square root of the variance therefore

Hence the standard deviation is mean of x is 68.75.

iii) To find

Using the dstribution of mean of x which is found in 1

Using the standard probability table for standard normal above probability is found out.

iv) Even if the monthly salaries were not normally distributed the above result would hold because the sample size is 64 which is quite high and becauseof large sample size any distribution converges to normal distribution.

So even if it was not normal because of large sample size we could easily approximate it to normal.