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 = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (P ≤ .025) in relation to the Z score.
To decide if each sample represents the population of average bowlers, we can conduct a hypothesis test using the criterion of p = .05.
Step 1: State the null and alternative hypotheses:
Null hypothesis (H0): The sample does represent the population of average bowlers.
Alternative hypothesis (Ha): The sample does not represent the population of average bowlers.
Step 2: Determine the test statistic and critical value:
Since the population standard deviation is known (6), we can use a Z-test to compare the sample means to the population mean.
The test statistic formula for the Z-test is:
Z = (x̄ - μ) / (σ / √n)
Where:
x̄ is the sample mean,
μ is the population mean,
σ is the population standard deviation, and
n is the sample size.
The critical value in a two-tailed test with a significance level of 0.05 is ±1.96 (obtained from the Z-table).
Step 3: Calculate the test statistic:
For the first sample at Fred's Bowling Alley:
x̄1 = 26, μ = 24, σ = 6, and n = 30.
Using the formula:
Z1 = (26 - 24) / (6 / √30)
Z1 = 2 / (6 / √30)
Z1 ≈ 0.816
For the second sample at Ethel's Bowling Alley:
x̄2 = 18, μ = 24, σ = 6, and n = 30.
Using the formula:
Z2 = (18 - 24) / (6 / √30)
Z2 = -6 / (6 / √30)
Z2 ≈ -3.09
Step 4: Compare the test statistics to the critical value:
In a two-tailed test, if the absolute value of the test statistic is higher than the critical value (1.96), we reject the null hypothesis.
|Z1| ≈ 0.816 < 1.96, so we fail to reject the null hypothesis for the first sample.
|Z2| ≈ 3.09 > 1.96, so we reject the null hypothesis for the second sample.
Step 5: Interpret the results:
Based on the hypothesis test, the first sample from Fred's Bowling Alley does represent the population of average bowlers. However, the second sample from Ethel's Bowling Alley does not represent the population of average bowlers.
In conclusion, the first sample represents the population, while the second sample does not.
To determine if each sample represents the population of average bowlers, we can conduct hypothesis tests and examine the p-values.
Let's start by stating the null and alternative hypotheses for each sample:
Sample from Fred's Bowling Alley:
Null Hypothesis (H0): The sample represents the population of average bowlers (µ = 24).
Alternative Hypothesis (HA): The sample does not represent the population of average bowlers (µ ≠ 24).
Sample from Ethel's Bowling Alley:
Null Hypothesis (H0): The sample represents the population of average bowlers (µ = 24).
Alternative Hypothesis (HA): The sample does not represent the population of average bowlers (µ ≠ 24).
Next, we can calculate the test statistics and p-values for each sample.
For the sample from Fred's Bowling Alley:
1. Calculate the test statistic (z-score) using the formula: z = (sample mean - population mean) / (population standard deviation / √sample size)
z = (26 - 24) / (6 / √30)
2. Refer to the standard normal distribution table or use statistical software to find the p-value associated with the calculated z-score.
For the sample from Ethel's Bowling Alley:
1. Calculate the test statistic (z-score) using the formula: z = (sample mean - population mean) / (population standard deviation / √sample size)
z = (18 - 24) / (6 / √30)
2. Refer to the standard normal distribution table or use statistical software to find the p-value associated with the calculated z-score.
Once we have the p-values, we can compare them to the significance level (α = 0.05). If the p-value is less than or equal to 0.05, we reject the null hypothesis and conclude that the sample does not represent the population of average bowlers. If the p-value is greater than 0.05, we fail to reject the null hypothesis, indicating that the sample represents the population of average bowlers.
Please note that this explanation assumes a two-tailed test since we are considering both tails of the sampling distribution.