a Rollercoaster's auditors estimate that the average daily loss from those illegally riding without tickets is at lease (greater or equal) $295, but wants to determine the accuracy of this statistic. The company researcher takes a random sample of losses over 64 days and finds that x=$300 and s=$15.

a) Test at a=0.05 Step 1: Hypothesis

Step 2: Critical value and acceptance-range

Step3: Compute test-value(calculated z)

Step 4: Accept or reject (ALWAYS EXPLAIN WHY YOU ACCEPT OR REJECT)

b) Construct a 94% confidence interval of losses
Note: a and b are independent.

a) Test at a = 0.05:

Step 1: Hypothesis
Null Hypothesis (H0): The average daily loss is $295.
Alternative Hypothesis (H1): The average daily loss is greater than $295.

Step 2: Critical value and acceptance-range
Since the alternative hypothesis is one-sided (greater than), we use the right-tailed z-test. The critical value for a = 0.05 can be found using a standard normal distribution table or calculator. Since a = 0.05 is a common significance level, the critical value for a one-tailed test is 1.645.

Step 3: Compute test-value (calculated z)
We can calculate the test-value or calculated z-score using the formula:
z = (x - μ) / (σ / √n)

Where:
x = sample mean = $300
μ = population mean (hypothesized) = $295
σ = population standard deviation = $15
n = sample size = 64

Using the given values, we can calculate the z-score:
z = (300 - 295) / (15 / √64) = 5 / (15 / 8) = 1.3333

Step 4: Accept or reject
Since the calculated z-score (1.3333) is less than the critical value (1.645), we do not reject the null hypothesis. Therefore, there is not enough evidence to conclude that the average daily loss is greater than $295 at a significance level of 0.05.

b) Construct a 94% confidence interval of losses:
To construct a confidence interval, we can use the formula:
CI = x ± (z * σ / √n)

Where:
x = sample mean = $300
z = z-score for a 94% confidence level (which can be found using a standard normal distribution table or calculator; it is approximately 1.8808)
σ = population standard deviation = $15
n = sample size = 64

Substituting the values:
CI = 300 ± (1.8808 * 15 / √64)

Calculating the confidence interval:
CI = 300 ± (1.8808 * 15 / 8) = 300 ± (35.641 / 8) = 300 ± 4.4551

The 94% confidence interval for the average daily loss is ($295.5449, $304.4551).