You measure 64 children, obtaining a X of 57.28. Slug says that bc this X is so close to the mean of 56, this sample could hardly be considered gifted .
A. Perform the appropriate statical procedure to determine whether he is correct.
B. In what percentage of the top scores is this sample mean?
Need standard deviation (SD) value.
Z = (score-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 Z score.
To determine whether the sample can be considered gifted or not, we need to perform a statistical procedure. In this case, since we have a sample mean and want to compare it with a population mean, we can use a one-sample t-test.
A. Performing a one-sample t-test:
1. Define the null hypothesis (H₀): The sample mean is equal to the population mean.
2. Define the alternative hypothesis (H₁): The sample mean is not equal to the population mean.
3. Set the significance level (α): This is the threshold we use to determine whether the results are statistically significant. Common values for α are 0.05 or 0.01, depending on the desired level of confidence.
4. Calculate the t-value: The t-value is obtained by subtracting the population mean from the sample mean and dividing it by the standard deviation of the sample divided by the square root of the sample size.
t = (X - μ) / (s / √n)
To calculate the t-value, we need to know the standard deviation of the sample, which is not provided in your question. If you have that information, you can plug in the values to calculate the t-value. Once you have the t-value, you can use a t-table or statistical software to determine the p-value.
B. To determine the percentage of top scores that the sample mean represents:
1. Calculate the z-score of the sample mean: The z-score tells us how many standard deviations the sample mean is away from the population mean.
z = (X - μ) / σ
Again, we need to know the standard deviation (σ) of the population to calculate the z-score. Once you have the z-score, you can use a standard normal distribution table (also known as a z-table) or statistical software to determine the percentage of top scores that fall below the sample mean.