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Sampling distribution of the mean

Understand the sample mean and sample proportion as random variables; apply the normal approximation to the binomial distribution; use the sampling distribution of Xˉ\bar{X} with mean μ\mu and standard error σ/n\sigma/\sqrt{n}.

Worked examples

Standard error of the sample mean

Straightforward

Problem

Random samples of size n=100n = 100 are taken from a population with μ=60\mu = 60 and σ=15\sigma = 15. Find the standard error of Xˉ\bar{X} and describe the distribution of Xˉ\bar{X}.

Normal approximation to the binomial

Moderate

Problem

X∼B(100,0.4)X \sim B(100, 0.4). Using the normal approximation, find P(X<35)P(X < 35).

Using the empirical rule

Challenging

Problem

X∼B(400,0.5)X \sim B(400, 0.5) is approximated by N(200,100)N(200, 100) (SD = 10). Use the empirical rule to find P(180≤X≤220)P(180 \leq X \leq 220).

Practise

Q1·Straightforward
X∼B(25,0.6)X \sim B(25, 0.6). When the binomial distribution is approximated by a normal distribution, what is the mean of the approximating normal distribution?
Q2·Straightforward
X∼B(25,0.5)X \sim B(25, 0.5). When the binomial distribution is approximated by a normal distribution, what is the standard deviation of the approximating normal distribution?
Q3·Straightforward
A random sample of size n=64n = 64 is taken from a population with standard deviation σ=8\sigma = 8. Find the standard error of the sample mean Xˉ\bar{X}.
Q4·Straightforward
X∼B(100,0.5)X \sim B(100, 0.5) is approximated by N(50,25)N(50, 25) (mean 50, variance 25). Find the zz-score corresponding to X=60X = 60.
Q5·Moderate
A random sample of n=100n = 100 observations is drawn from a population with mean μ=50\mu = 50 and standard deviation σ=20\sigma = 20. Find the standard error of the sample mean Xˉ\bar{X}.
Q6·Moderate
X∼B(400,0.5)X \sim B(400, 0.5) is approximated by N(200,100)N(200, 100) (mean 200, standard deviation 10). Using the empirical rule (68–95–99.7 rule), find P(190<X<210)P(190 < X < 210).
Q7·Moderate
A random sample of size n=36n = 36 is taken from a population with mean μ=80\mu = 80 and standard deviation σ=12\sigma = 12. The sample mean Xˉ\bar{X} is approximately N(80,4)N(80, 4) (mean 80, standard deviation 2). Using the empirical rule, find P(Xˉ>82)P(\bar{X} > 82).
Q8·Moderate
The true proportion of voters who support a candidate is p=0.3p = 0.3. A random sample of n=100n = 100 voters is surveyed. The standard error of the sample proportion p^\hat{p} is SE(p^)=p(1−p)/n\text{SE}(\hat{p}) = \sqrt{p(1-p)/n}. Find SE(p^)\text{SE}(\hat{p}) to 4 decimal places.
Q9·Moderate
X∼B(196,0.5)X \sim B(196, 0.5) is approximated by N(98,49)N(98, 49) (mean 98, standard deviation 7). Using the empirical rule, find P(X<105)P(X < 105).
Q10·Challenging
Samples of size n=25n = 25 are repeatedly taken from a population with mean μ=70\mu = 70 and standard deviation σ=10\sigma = 10. The sample mean Xˉ\bar{X} is approximately normal with standard error SE=2\text{SE} = 2. Using the empirical rule, find P(∣Xˉ−70∣>4)P(|\bar{X} - 70| > 4).
Q11·Challenging
For X∼B(n,0.4)X \sim B(n, 0.4), the normal approximation is valid when both np≥5np \geq 5 and n(1−p)≥5n(1-p) \geq 5. Find the minimum value of nn for which both conditions are satisfied.
Q12·Challenging
In a large population, the true proportion who prefer a new product is p=0.5p = 0.5. A random sample of n=100n = 100 people is surveyed, and the sample proportion p^\hat{p} has standard error SE=0.05\text{SE} = 0.05. Using the empirical rule, find P(0.40≤p^≤0.60)P(0.40 \leq \hat{p} \leq 0.60).