Question

To investigate if autism is marked by different brain growth patterns in early life, studies have tried to link brain size in infants and toddlers to autism. Suppose the whole-brain volume in non-autistic toddlers is known to be 1200 milliliters, on average. One study based on a sample size of 25 autistic toddlers had a sample mean volume of 1280 ml with a standard deviation of 230 ml.
a. Calculate a 95% confidence interval for the whole-brain volume of autistic toddlers. Given: ?"#$%& ∗ =2.064

b. Do the results from this study suggest that autistic toddlers have the same whole-brain volume as non-autistic toddlers, on average? Explain your answer.
c. If you used the same data from this study and calculated a 90% confidence interval, would it be wider or narrower than the 95% interval?
d. If the data from this study had the same mean and standard deviation, but was from a sample of size 15 instead of 25, would the 95% confidence have been wider or narrower?

Answer 1

SOLUTION A: SAMPLE MEAN = 1280 ML
t critical = 2.06
s_M = √(230²/25) = 46
μ = M ± t(s_M)
μ = 1280 ± 2.06*46
μ = 1280 ± 94.76

95% CI [1185.24, 1374.76].

You can be 95% confident that the population mean (μ) falls between 1185.24 and 1374.76.

Solution b: Since 95% confidence interval does contain true population mean 1200 therefore NOT significant and hence we can conclude that the results from this study suggest that autistic toddlers have the same whole-brain volume as non-autistic toddlers, on average.

Solution C: Width of 95% CI= 1374.76-1185.24= 189.52

t critical for 90% CI= 1.71
s_M = √(230²/25) = 46

μ = M ± t(s_M)
μ = 1280 ± 1.71*46
μ = 1280 ± 78.7

90% CI [1201.3, 1358.7].

You can be 90% confident that the population mean (μ) falls between 1201.3 and 1358.7.

WIDTH OF 90% CI= 1358.7-1201.3= 157.4

Width of 95% CI is more than 90% CI

d) For n=15

s_M = √(230²/15) = 59.39
μ = M ± t(s_M)
μ = 1280 ± 2.14*59.39
μ = 1280 ± 127.09

95% CI [1152.91, 1407.09].

You can be 95% confident that the population mean (μ) falls between 1152.63 and 1407.09.

Width of 95%CI= 1407.09-1152.91=254.18

Width of 95% CI for n=15 GREATER than the width of 95% CI for n=25