a. This is a one-tailed test because the researcher has a specific hypothesis that individuals who listen to classical music will score differently from the general population on the test of spatial ability.
b. H0 (null hypothesis): There is no difference in the mean score of individuals who listen to classical music compared to the general population on the test of spatial ability.
Ha (alternative hypothesis): Individuals who listen to classical music will score differently from the general population on the test of spatial ability.
c. We need to compute tobt using the given information. To calculate tobt, we will use the formula:
tobt = (x̄ - μ) / (s / sqrt(n))
Where:
x̄ is the sample mean (average score of individuals who listen to classical music) = (52 + 59 + 63 + 65 + 58 + 55 + 62 + 63 + 53 + 59 + 57 + 61 + 60 + 59) / 14 = 59.1429
μ is the population mean (given in the question) = 58
s is the sample standard deviation, which can be calculated using the formula = sqrt(sum((xi - x̄)^2) / (n-1)) = 3.3889
n is the sample size = 14
Now we can substitute the values into the formula:
tobt = (59.1429 - 58) / (3.3889 / sqrt(14))
tobt ≈ 0.2503
d. To determine tcv (critical value), we need to look up the value from the t-distribution table. The degrees of freedom for this test are (n-1) = (14-1) = 13. Assuming a significance level of 0.05 (α = 0.05), for a one-tailed test, the critical value is approximately 1.771.
tcv ≈ 1.771
e. To determine if H0 should be rejected, we compare tobt with tcv. If tobt is greater than tcv, we reject H0; otherwise, we fail to reject H0.
In this case, tobt (0.2503) is smaller than tcv (1.771), so we fail to reject H0. Therefore, there is no sufficient evidence to conclude that individuals who listen to classical music score differently from the general population on the test of spatial ability.
f. To determine the 95% confidence interval for the population mean based on the sample mean, we can use the formula:
CI = x̄ ± (tcv * (s / sqrt(n)))
Substituting the values:
CI = 59.1429 ± (1.771 * (3.3889 / sqrt(14)))
CI ≈ 59.1429 ± 3.4647
CI ≈ (55.6782, 62.6076)
Therefore, the 95% confidence interval for the population mean, based on the sample mean, is approximately 55.6782 to 62.6076. This means that we can be 95% confident that the true population mean lies within this interval.