A final exam in math 160 has a mean of 73 with a standard deviation 7.73. Assume that a random sample of 24 students is selected and the test score of the sample is computed. Assuming the scores are normally distributed, what percentage of the samplerans are less than 70?
First, we need to calculate the standard error of the mean:
SE = σ/√n = 7.73/√24 ≈ 1.58
Next, we can standardize the sample mean using the formula:
z = (x̄ - μ)/SE
where x̄ is the sample mean, μ is the population mean (73), and SE is the standard error of the mean. Substituting the given values, we get:
z = (70 - 73)/1.58 ≈ -1.90
Using a standard normal distribution table or calculator, we can find that the percentage of samples with a mean less than 70 is approximately 2.28%. Therefore, we can conclude that about 2.28% of samples of size 24 would have a mean score less than 70 on the math 160 final exam, assuming the scores are normally distributed.
To find the percentage of sample scores that are less than 70, we need to calculate the z-score and then use a z-table to find the corresponding percentage.
Step 1: Calculate the z-score using the formula:
z = (x - μ) / σ
where x is the value we want to find the percentage for (70), μ is the mean (73), and σ is the standard deviation (7.73).
z = (70 - 73) / 7.73
Step 2: Calculate the z-score using a calculator or spreadsheet:
z ≈ -0.388
Step 3: Use a z-table to find the percentage corresponding to the z-score. Since we want to find the percentage of scores less than 70, we need to find the area to the left of the z-score (-0.388) in the z-table.
Looking up z = -0.388 in a standard normal distribution table, we find that the area to the left of -0.388 is approximately 0.35.
Step 4: Convert the decimal to a percentage:
0.35 * 100 = 35%
Therefore, approximately 35% of the sample scores will be less than 70.