2. Betty, an employee of Shining Sun Daycare Center, read an article in Healthy Child Magazine saying that the average 3-year-old child is 37 in. tall. Betty works with 3-year-olds at Shining Sun, so later that week, she measured the height of each child who had just turned or was about to turn 3 years old. Here are their heights in inches: 41, 40, 36, 42, 39, 38, 33, 44, 39, 41.
a. State the nondirectional hypothesis.
b. Determine the critical t for α = .05.
c. Calculate t. Show your calculations.
d. Is the height of 3-year-olds in Shining Sun Daycare Center significantly different from the height given in the magazine?
a. The nondirectional hypothesis is that the average height of 3-year-olds at Shining Sun Daycare Center is different from the average height reported in Healthy Child Magazine.
b. To determine the critical t for α = .05, we need to consult the t-distribution table based on the degrees of freedom (n-1) and the desired significance level (α). Since we don't have the sample size (n) provided in the question, we cannot calculate the degrees of freedom or the critical t value.
c. To calculate t, we need the sample mean, sample standard deviation, and the sample size. The given heights are: 41, 40, 36, 42, 39, 38, 33, 44, 39, 41.
First, we calculate the sample mean:
Mean (x̄) = (41 + 40 + 36 + 42 + 39 + 38 + 33 + 44 + 39 + 41) / 10
Mean (x̄) = 393 / 10
Mean (x̄) = 39.3 inches
Next, we calculate the sample standard deviation:
Step 1: Find the difference between each value and the sample mean.
(41 - 39.3), (40 - 39.3), (36 - 39.3), (42 - 39.3), (39 - 39.3), (38 - 39.3), (33 - 39.3), (44 - 39.3), (39 - 39.3), (41 - 39.3)
Step 2: Square each difference.
(1.7)^2, (0.7)^2, (-3.3)^2, (2.7)^2, (-0.3)^2, (-1.3)^2, (-6.3)^2, (4.3)^2, (-0.3)^2, (1.7)^2
Step 3: Sum up the squared differences.
1.7^2 + 0.7^2 + (-3.3)^2 + 2.7^2 + (-0.3)^2 + (-1.3)^2 + (-6.3)^2 + 4.3^2 + (-0.3)^2 + 1.7^2 = 94.3
Step 4: Divide the sum by (n-1) to calculate the sample variance.
Sample Variance (s^2) = 94.3 / (10 - 1) = 94.3 / 9 = 10.48
Step 5: Take the square root of the sample variance to get the sample standard deviation.
Sample Standard Deviation (s) = sqrt(10.48) = 3.23
Now that we have the sample mean (x̄) = 39.3 inches and the sample standard deviation (s) = 3.23 inches, we can proceed to calculate t.
d. We cannot answer part d without knowing the critical t value from part b and the sample size (n). Since those details are missing, we cannot determine if the height of 3-year-olds in Shining Sun Daycare Center is significantly different from the height given in the magazine.