A study of 40 bowlers showed that their average score was 186. The standard deviation of the population is 6. Find the 95% confidence of the mean score for all bowlers.
95% = mean ± 1.96 SEm
SEm = SD/√n
PA help
To find the 95% confidence interval for the mean score for all bowlers, we will use the formula:
Confidence interval = mean ± (critical value) * (standard deviation / √n)
where:
- mean is the sample mean (given as 186)
- critical value is obtained from the Z-table for a 95% confidence level (which corresponds to a 2-sided alpha of 0.05)
- standard deviation is the population standard deviation (given as 6)
- n is the sample size (given as 40)
Now, let's calculate the confidence interval step-by-step.
Step 1: Find the critical value
Since we want a 95% confidence interval, the alpha level is 0.05 (2-sided test). Half of this alpha value is 0.025. Using the Z-table or a calculator, we can find that the critical value for a 2-sided 95% confidence level is approximately 1.96.
Step 2: Calculate the standard error of the mean (SEM)
The standard error of the mean (SEM) measures the variability of sample means. It is calculated as the standard deviation divided by the square root of the sample size.
SEM = standard deviation / √n
In this case, the standard deviation is given as 6, and the sample size is given as 40.
SEM = 6 / √40 ≈ 0.948
Step 3: Calculate the margin of error
The margin of error represents the range around the sample mean within which the population mean is likely to fall.
Margin of error = critical value * SEM
Margin of error = 1.96 * 0.948 ≈ 1.859
Step 4: Calculate the confidence interval
The confidence interval is calculated by subtracting and adding the margin of error to the sample mean.
Confidence interval = mean ± margin of error
Confidence interval = 186 ± 1.859
Step 5: Write the final answer
The 95% confidence interval for the mean score for all bowlers is approximately 184.141 to 187.859 (rounded to three decimal places).
Therefore, we can be 95% confident that the true population mean score for all bowlers lies within the range of 184.141 to 187.859.
To find the 95% confidence interval for the mean score of all bowlers, we can use the formula:
Confidence interval = X̄ ± (Z * (σ / √n))
Where:
- X̄ is the sample mean (186 in this case).
- Z is the Z-score corresponding to the desired confidence level (95% in this case). The Z-score for a 95% confidence level is approximately 1.96.
- σ is the population standard deviation (6 in this case).
- n is the sample size (40 in this case).
Let's calculate the confidence interval step by step:
Step 1: Plug in the known values into the formula.
Confidence interval = 186 ± (1.96 * (6 / √40))
Step 2: Simplify the formula.
Confidence interval = 186 ± (1.96 * 0.948)
Step 3: Perform the calculations.
Confidence interval = 186 ± 1.85808
Step 4: Calculate the lower and upper limits of the confidence interval.
Lower limit = 186 - 1.85808 ≈ 184.14192
Upper limit = 186 + 1.85808 ≈ 187.85808
Step 5: Interpret the results.
The 95% confidence interval for the mean score of all bowlers is approximately (184.14, 187.86).
Therefore, we are 95% confident that the true mean score for all bowlers falls within this interval.