An electric car manufacturer specifies that the car battery of their long range model has a capacity of 65 kilowatt-hours (kWh). They define the battery to be defective if the capacity is less than 64.5 kWh. Past experience suggests that the population mean battery capacity is 66.24 kWh with population standard deviation of 2.16kW. They are willing to assume that the battery capacities are normally distributed. It can also be assumed that the battery capacities of different cars are statistically independent.

(a) What is the probability that a particular battery is defective?

(b) The company quality control manager is confident that the assumption of normality of battery capacities is justified, because of the Central Limit Theorem. Is she correct?

(c)A batch of 8 cars is delivered to a particular store. What is the probability that two or more of these cars have defective batteries?

(d)) In a production batch of 204 cars, what is the the probability that 47 or more cars have defective batteries? Calculate this probability using the normal approximation to the binomial distribution, with continuity correction.

(e)To check their assumptions, they obtain data for a sample of 6 cars. The sample mean of these cars' battery capacities is 67.5 kWh with a sample standard deviation of 1.52 kWh. Obtain a 80% confidence interval for the population mean battery capacity based on these observations. Do not assume that the population standard deviation is known when calculating this confidence interval.