Chicken Delight claims that 90 percent of its orders are delivered within 10 minutes of the time the order is placed. A sample of 70 orders revealed that 60 were delivered within the promised time. At the .02 significance level, can we conclude that less than 90 percent of the orders are delivered in less than 10 minutes?
Compute the value of the test statistic. (Negative amount should be indicated by a minus sign. Round sample proportion to 2 decimal places. Round your answer to 2 decimal places.)
http://www.mdc.edu/asa/documents/competencies/pdf/QMB2100.pdf
See the Related Questions below.
To compute the value of the test statistic, we can follow these steps:
Step 1: State the hypotheses.
- Null Hypothesis (H0): p = 0.9 (90% of orders are delivered within 10 minutes)
- Alternative Hypothesis (Ha): p < 0.9 (less than 90% of orders are delivered within 10 minutes)
Step 2: Set the significance level.
The significance level (α) is given as 0.02.
Step 3: Calculate the test statistic.
In this case, we will use the z-test since we have a large sample size and are testing a proportion.
The formula for the z-test statistic is:
z = (p̂ - p) / sqrt( (p * (1 - p)) / n )
where:
p̂ = sample proportion
p = hypothesized proportion
n = sample size
We are given:
p̂ = 60/70 ≈ 0.86 (proportion of orders delivered within 10 minutes in the sample)
p = 0.9 (hypothesized proportion)
n = 70 (sample size)
Substituting the values into the formula:
z = (0.86 - 0.9) / sqrt( (0.9 * (1 - 0.9)) / 70 )
Calculating this expression gives us:
z ≈ -3.53 (rounded to 2 decimal places)
So the value of the test statistic is approximately -3.53.
To compute the test statistic in this hypothesis test, we need to follow these steps:
Step 1: State the null hypothesis (H0) and the alternative hypothesis (Ha).
H0: p = 0.90 (90 percent of orders are delivered within 10 minutes)
Ha: p < 0.90 (less than 90 percent of orders are delivered within 10 minutes)
Step 2: Calculate the sample proportion (p-hat):
In this case, the sample proportion is calculated as the number of orders delivered within 10 minutes divided by the total number of orders in the sample:
p-hat = 60/70 ≈ 0.8571 (rounded to 4 decimal places)
Step 3: Calculate the test statistic (z-score):
The formula for the test statistic (z-score) when testing a proportion is:
z = (p-hat - p) / √(p * (1 - p) / n)
where:
p-hat is the sample proportion,
p is the hypothesized population proportion (from the null hypothesis),
n is the sample size.
In this case, the formula becomes:
z = (0.8571 - 0.90) / √(0.90 * (1 - 0.90) / 70)
Calculating the values:
z = (-0.0429) / √(0.90 * 0.10 / 70)
z = -0.0429 / √(0.09 / 70)
z = -0.0429 / √(0.0012857142857142857)
z ≈ -2.50 (rounded to 2 decimal places)
Therefore, the test statistic is approximately -2.50.
Note: The test statistic represents the number of standard deviations the sample proportion is away from the hypothesized population proportion. In this case, it indicates how far the sample proportion of orders delivered within 10 minutes (0.8571) is from the hypothesized proportion of 0.90.