Chicken Delight claims that 91% of its orders are delivered within 10 minutes of the time the order is placed. A sample of 90 orders revealed that 75 were delivered within the promised time. At the 0.10 significance level, can we conclude that less than 91% of the orders are delivered in less than 10 minutes?
What is the decision rule? (Reject Ho if z <)
Compute the value of the test statistic.
To determine whether we can conclude that less than 91% of the orders are delivered within 10 minutes, we need to perform a hypothesis test.
Let's set up the null and alternative hypotheses:
- Null hypothesis (H0): The proportion of orders delivered within 10 minutes is equal to 91%.
- Alternative hypothesis (Ha): The proportion of orders delivered within 10 minutes is less than 91%.
Now, we need to determine the decision rule at a 0.10 significance level. The significance level, denoted by α, represents the probability of rejecting the null hypothesis when it is actually true. In this case, α = 0.10.
Since we are testing whether the proportion of orders delivered within 10 minutes is less than 91%, this is a one-tailed test. Therefore, we'll use the z-test and the critical z-value associated with a 0.10 significance level.
To find the critical value, we can use a z-table or a statistical software. The z-table shows that the critical z-value for a one-tailed test at a significance level of 0.10 is approximately -1.28. So, the decision rule is:
- Reject H0 if z < -1.28.
Next, let's compute the value of the test statistic (z-score).
We have a sample of 90 orders, and out of those, 75 were delivered within the promised time. To calculate the test statistic (z-score) for a proportion, we can use the formula:
z = (p̂ - p) / √(p(1-p)/n)
Where:
- p̂ is the sample proportion (number of orders delivered within 10 minutes / total number of orders)
- p is the hypothesized proportion (91% or 0.91)
- n is the sample size (90)
Let's plug in the values:
p̂ = 75/90 = 0.8333 (rounded to four decimal places)
p = 0.91
n = 90
Now, we can calculate the test statistic:
z = (0.8333 - 0.91) / √(0.91 * (1 - 0.91) / 90)
Calculating the test statistic value:
z = -2.1626 (rounded to four decimal places)
So, the calculated test statistic (z-score) is -2.1626.