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Practice problems
Normal distribution
Please determine whether the variable was normally distributed based on each individual statistic.
Statistic and result Was the data normally distributed?
Shapiro-Wilk test, p = 0.
Shapiro-Wilk test, p = 0.
Shapiro-Wilk test, p = 0.
Skewness = - 2.
Skewness = - 0.
Skewness = 1.
Skewness = 0.
Kurtosis = - 1.
Kurtosis = 2.
Kurtosis = - 0.
Kurtosis = 0.
z-test for one sample mean
Problem 1. (two-tailed analysis, alpha = 0.05)
Research hypothesis: Soccer players will exhibit a higher resting energy expenditure than the
Canadian adult population.
Data:
Sample Population
𝑋" = 1950 kcal/day μ = 1690 kcal/day
SD = 470 kcal/day σ = 4 80 kcal/day
n = 12
Problem 2. (two-tailed analysis, alpha = 0.05)
Research hypothesis: The varsity volleyball players will achieve a higher squat jump height than the
Canadian adult population.
Data:
Sample Population
𝑋" = 29 cm μ = 23 cm
SD = 5 cm σ = 10 cm
n = 35
Problem 3. (two-tailed analysis, alpha = 0.05)
Research hypothesis: The swimmers will exhibit higher handgrip strength scores than the Canadian
adult population.
Data:
Sample Population
𝑋" = 35 kg μ = 39 kg
SD = 13 kg σ = 14 kg
n = 65
Solved problems
Normal distribution
Please determine whether the variable was normally distributed based on each individual statistic.
Statistic and result Was the data normally distributed?
Shapiro-Wilk test, p = 0.51 Yes, when p > 0.05, the data was
normally distributed
Shapiro-Wilk test, p = 0.03 No, when p < 0.05, the data was not
normally distributed
Shapiro-Wilk test, p = 0.82 Yes, when p > 0.05, the data was
normally distributed
Skewness = - 2.3 No, the degree of left asymmetry is of
concern (<-1)
Skewness = - 0.3 Yes, the degree of left asymmetry is
not of concern (>-1 but <1)
Skewness = 1.2 No, the degree of right asymmetry is
of concern (>1)
Skewness = 0.6 Yes, the degree of right asymmetry is
not of concern (>-1 but <1)
Kurtosis = - 1.3 No, the degree of lightness on the tails
is of concern (<-1)
Kurtosis = 2.2 No, the degree of heaviness on the
tails is of concern (>1)
Kurtosis = - 0.1 Yes, the degree of lightness on the
tails is not of concern (>-1 but <1)
Kurtosis = 0.2 Yes, the degree of heaviness on the
tails is not of concern (>-1 but <1)
z-test for one sample mean
Problem 1. (two-tailed analysis, alpha = 0.05)
Research hypothesis: Soccer players will exhibit a higher resting energy expenditure than the
Canadian adult population.
Data:
Sample Population
𝑋" = 1950 kcal/day μ = 1690 kcal/day
SD = 470 kcal/day σ = 480 kcal/day
n = 12
1. H0: μ = 1690 kcal/day
H1: μ ≠1690 kcal/day
2. 2a. alpha = 0.05 (0.05 / 2 = 0.025)
2b. Two-tailed
2c. z-critical = ±1.
2d. If z is ≤-1.96 or ≥1.96, we reject the null hypothesis; otherwise, we do not reject the null
hypothesis
2e. Population standard error of the mean
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2 f. Because z (1.87) is not ≤ or ≥ the z-critical (±1.96), we do not reject the null hypothesis (p
3. The soccer players (n = 12) exhibited a similar resting energy expenditure (1950 ± 470
kcal/day) to the Canadian adult population (1690 ± 480 kcal/day) (z = 1.87, two-tailed p >
We can also say: The soccer players (n = 12) exhibited a non-statistically significantly
digerent resting energy expenditure (1950 ± 470 kcal/day) compared to the Canadian adult
population (1690 ± 480 kcal/day) (z = 1.87, two-tailed p > 0.05).
4. The results do not support the research hypothesis. Soccer players' resting energy
expenditure is similar to that of the Canadian adult population.
Commented [SG1]: This one is technically more
appropriate because we mention the statistical
significance. Still, both ways are correct.
Problem 3. (two-tailed analysis, alpha = 0.05)
Research hypothesis: The swimmers will exhibit higher handgrip strength scores than the Canadian
adult population.
Data:
Sample Population
𝑋" = 3 5 kg
μ = 39 kg
SD = 13 kg σ = 14 kg
n = 65
1. H0: μ = 39 kg
H1: μ ≠ 39 kg
2. 2a. alpha = 0.05 (0.05 / 2 = 0.025)
2b. Two-tailed
2c. z-critical = ±1.
2d. If z is ≤-1.96 or ≥1.96, we reject the null hypothesis; otherwise, we do not reject the null
hypothesis
2e. Population standard error of the mean
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z-statistic
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2f. Because z (-2.30) is less than the z-critical (±1.96), we reject the null hypothesis (p <
p < 0.05 - > z-critical = ±1.96 (because - 2.30 < - 1.96, but not < the other critical values, p <
0.0 5 is the most precise p-value)
p < 0.01 - > z-critical = ±2.
p < 0.001 - > z-critical = ±3.
3. The swimmers (n = 65 ) exhibited a statistically significantly lower handgrip strength ( 35 ± 13
kg) than the Canadian adult population ( 39 ± 14 kg) (z = - 2.30, two-tailed p < 0. 05 ).
4. The results do not support the research hypothesis. Based on handgrip strength, the
swimmers are not stronger than the Canadian adult population but weaker.
t-test for one sample mean
Problem 1. (two-tailed analysis, alpha = 0.05)
Research hypothesis: Patients with heart disease will show fewer steps during the day than the
international recommendation.
Data:
Sample Population
𝑋" = 7550 steps/day μ = 10,000 steps/day
SD = 1660 steps/day
n = 25
1. H0: μ = 10,000 steps/day
H1: μ ≠ 10,000 steps/day
2. 2a. alpha = 0.
2b. Two-tailed
2c. Df = 25 – 1 = 24
2 d. t-critical = ±2.
2 e. If t is ≤-2.064 or ≥2.064, we reject the null hypothesis; otherwise, we do not reject the
null hypothesis
2 f. Standard error of the mean
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t-statistic
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2 g. Because t (-7.380) is less than the t-critical (±2.064), we reject the null hypothesis (p <
p < 0.05 - > t-critical = ±2.
p < 0.01 - > t-critical = ±2. 797
p < 0.001 - > t-critical = ±3. 745 (because - 7.380 < - 3. 745 , p < 0.001 is the most precise p-
value)
Because t (-7.380) is less than the t-critical (±2.064), we reject the null hypothesis (p <
3. The patients with heart disease (n = 25 ) exhibited statistically significantly fewer steps per
day ( 7550 ± 1660 ) than the international recommendation ( 10 ,000 steps/day) (t (24) = -
7.380, two-tailed p < 0. 001 ).
4. The results support the research hypothesis. On average, patients with heart disease do not
meet the international recommendation for steps a day, which means they are less
physically active than this recommendation.
Problem 3. (two-tailed analysis, alpha = 0.05)
Research hypothesis: The climbers (bouldering) will show greater handgrip strength than the
average Canadian adult population.
Data:
Sample Population
𝑋" = 53 kg
μ = 39 kg
SD = 21 kg
n = 13
1. H0: μ = 39 kg
H1: μ ≠ 39 kg
2. 2a. alpha = 0.
2b. Two-tailed
2c. Df = 13 – 1 = 12
2 d. t-critical = ±2.
2 e. If t is ≤-2.179 or ≥2.179, we reject the null hypothesis; otherwise, we do not reject the
null hypothesis
2 f. Standard error of the mean
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t-statistic
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t =
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2 g. Because t (2. 405 ) is greater than the t-critical (±2. 179 ), we reject the null hypothesis (p <
p < 0.05 - > t-critical = ±2. 179 (because 2. 405 > 2. 179 , but not > the other critical values, p <
0.05 is the most precise p-value)
p < 0.01 - > t-critical = ±3.
p < 0.001 - > t-critical = ±4.
3. The climbers (n = 13 ) exhibited statistically significantly higher handgrip strength scores ( 53
± 21 kg) than the average Canadian adult population (39 kg) (t ( 12 ) = 2. 405 , two-tailed p <
4. The results support the research hypothesis. The climbers have stronger handgrips than the
average Canadian adult population.
