Question

Hoping to lure more shoppers downtown, a city builds a new public parking garage in the central business district. The city plans to pay for the structure through parking fees. For a random sample of 40 weekdays, daily parking income averaged 150 dollars with a standard deviation of 23 dollars. A histogram showed a right-skewed shape. (a) Are the conditions met for doing a confidence interval? (Check all that apply.) None of the conditions are met. The Nearly Normal condition is met because the population is bell-shaped. The Nearly Normal condition is met because the sampling distribution of x-bar is bell-shaped. The sample was randomly chosen. The Nearly Normal condition is met because the data are bell-shaped. (b) Find a 98% confidence interval for the mean daily income generated from parking fees (use 4 decimal places): ($ , $ ) (c) Explain what this confidence interval means: There is 98% confidence that the interval contains the true mean daily income. There is 98% confidence that a randomly chosen day will have an income inside the interval. There is confidence that 98% of the days will have an income inside the interval. There is 98% confidence that the interval contains the sample mean daily income. (d) The consultant who advised the city on this project predicted that parking revenues would average $157 per day. Based on your confidence interval, what do you think of the consultant's prediction? Since the confidence interval the predicted value of $157, the consultant's claim plausible.

Answer 1

Assumptions

The data is randomly sampled.

The sample values are independent

The sample size must be sufficiently large. According to the Central Limit Theorem tells us that we can use a Normal model when the sample size is larger than 30 but here a histogram made by this 40 observations is a right skewed data. So the bell shaped curve of the sample is not achieved. But we can assume that the population mean has a bell shaped curve and almost symmetrical.So if you try to get more observations say 60 or 70 week days the sample data may appear almost symmetrical. If the population is very skewed, you will need a pretty large sample size to use the CLT, however if the population is unimodal and symmetric, even small samples are enough. so the answer will be The Nearly Normal condition is met because the population is bell-shaped.

Since the population is assumed to be normally distributed we can obtain the confidence interval

z value for 98% confidence is 2.326. Now the margin of error ,

The 98% CI is

There is confidence that 98% of the days will have an income inside the interval.

Since the confidence interval contains the predicted value of $157, the consultant's claim plausible.