**Question 1: Mean Temperature of Boston over the last 20 years.**

- The distribution is normal.

Reasons:

- The median, mean, and mode are almost equal (approximately 72)
- Exactly half of the values are to the right, and exactly half are on the left
- The curve is symmetric at the centre

- An outlier refers to an observation point, which is located far from other positions. The existence of an outlier may indicate an error in the experiment but may also be due to variability in measurements. In the present plot, there is no outlier since no single point is distant from other observations.
- The probability of the mean being above 76 in any July = 30%
- The probability of the mean being above 80 in any July = 0%

**Question 2**

The days with temperatures >=90 degree consecutively are Day5, Day6 and Day 7 Fahrenheit. For the 20 days represented in the data, this is the only heatwave.

Where P is the total number of probabilities.

**Question 3: Customer Survey**

The probability of sales made online

The probability of sales made physically

- Yes. The scenario suits the parameters for a binomial distribution since it fulfils the binomial distribution conditions as follows

- Sales made do not depend on each other.
- Only two possible outcomes; sales are either made online or physically.
- The number of trials is fixed at 12 sales

- The probability that on a given day, exactly 4 of the 12 sales are made online

If 4 sales are made online, 6 will be made physically. Therefore, there is no chance of making 12 sales on a given day = 0. The only possibility is that 10 sales can be made on the same day.

- 12 sales fewer than 6 are made online

Under this condition, the number of sales made can be 1, 2, 3, 4, or 5 (whole numbers between 0 and 6). Thus, P(1/0.4, 2/0.4, 3/0.4, 4/0.4, 5/0.4) = (2.5, 5, 7.5, 10, 12.5). Thus, the only case that satisfies the condition is 5, which means that P(of 12 sales fewer than 6 are made online) is 0.2 or 1/5.

- More than 8 are made online

Applying a similar argument as in (e) above, P(8/0.4,9/0.4,..) = (20,22.5,..), which means 8, 9, 10, 12, ….. Thus, the probability of more than 8 are made online is 1.

**Question 4**

Uber, a globally traded transport firm, is one of the businesses, which recently experienced challenges and scandals. Less than three years ago, the organization’s information system was hacked –enabling hackers to access critical data related to clients. In Uber’s case, probability and distribution analysis could help to estimate the probability of breach for their database applications. The company could use the approximations to determine vulnerable apps and assign risk ratings based on observable application features.

Suppose I were Uber’s business analyst, I would employ probability statistics to evaluate the firm’s performance based on its growth history and prospects, client perception, market position, and likely sales. Therefore, I would collect data related to all these variables and compare with competitors’. The case would be a binomial distribution since the assessment would involve particular historical and future events as well as their impacts and the likelihood of occurrence. The questions I would ask include

- What is the probability of Uber’s sales declining as a result of the event?
- What is the chance that the company will recover within the next five months?
- What is the probability that an intensive marketing campaign would restore customers’ confidence?

First, it is essential for the management to note the significance of emphasizing on client-related data in winning customers’ confidence. The transport market is highly competitive with competitors such as Taxify exploiting every single opportunity outdo their rivals. Therefore, organizations that guarantee safety to customers are more likely to win the game. In that regard, one of the most critical steps for the company to make its information system safe is to assess the probability of security risks occurring and this is where investing in probability analysis is vital. On the other hand, the evaluation would help the business to understand how various factors affect profitability.