On a test of motor coordination, the population of average bowlers has a mean score of 24, with a standard deviation of 6. A random sample of 30 bowlers at Fred's Bowling Alley has a sample mean of 26. A second random sample of 30 bowlers at Ethel's Bowling Alley has a mean of 18. Using the criterion of p = .05 and both tails of the sampling distribution, decide if each sample represents the population of average bowlers?
Z = (sample-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability
related to each Z score.
To determine if each sample represents the population of average bowlers, we can conduct hypothesis tests. The criterion of p = .05 suggests that we will use a significance level (alpha) of .05.
Let's start with the first sample at Fred's Bowling Alley:
1. Establish the null and alternative hypotheses:
- Null hypothesis (H0): The sample from Fred's Bowling Alley represents the population of average bowlers (mean = 24).
- Alternative hypothesis (H1): The sample from Fred's Bowling Alley does not represent the population of average bowlers (mean ≠ 24).
2. Set up the test statistic:
For a known population standard deviation and a sample mean, we can calculate the z-score using the formula:
z = (x - μ) / (σ / √n)
where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
In this case, x = 26, μ = 24, σ = 6, and n = 30.
3. Calculate the z-score:
z = (26 - 24) / (6 / √30)
4. Determine the critical z-values:
Since we want to consider both tails of the sampling distribution, we need to find the critical z-values that correspond to the chosen significance level of .05/2 = .025.
Using a standard normal distribution table or a calculator, we find that the critical z-values are approximately -1.96 and 1.96.
5. Compare the calculated z-score with the critical z-values:
If the calculated z-score falls between -1.96 and 1.96, we fail to reject the null hypothesis. Otherwise, we reject the null hypothesis and conclude that the sample does not represent the population of average bowlers.
Now, let's perform the same steps for the second sample at Ethel's Bowling Alley:
1. Null hypothesis (H0): The sample from Ethel's Bowling Alley represents the population of average bowlers (mean = 24).
Alternative hypothesis (H1): The sample from Ethel's Bowling Alley does not represent the population of average bowlers (mean ≠ 24).
2. Set up the test statistic:
x = 18, μ = 24, σ = 6, and n = 30.
3. Calculate the z-score:
z = (18 - 24) / (6 / √30)
4. Determine the critical z-values:
Using the significance level of .05/2 = .025, the critical z-values remain -1.96 and 1.96.
5. Compare the calculated z-score with the critical z-values:
Depending on where the calculated z-score falls, we either fail to reject the null hypothesis (if between -1.96 and 1.96) or reject the null hypothesis (if outside that range), indicating whether the sample represents the population of average bowlers.
By following these steps and comparing the calculated z-scores with the critical z-values, we can decide if each sample represents the population of average bowlers.