Q: A random sample of 49 transactions at a grocery store had a mean purchase of $$84 with a sample standard deviation of 14. $$Construct a 95% confidence interval for the mean purchase.

$$84 \pm t_{0.025,48} \times \frac{14}{\sqrt{49}} = 84 \pm 2.01 \times 2 = 84 \pm 4.02 = (79.98, 88.02)$$

Q: A survey of 300 customers found that 198 were satisfied with the company’s service. Construct a 90% confidence interval for the proportion of satisfied customers.

$$\hat{p} = 0.66,\ SE = \sqrt{0.66 \times 0.34 / 300} = 0.0275$$ $$0.66 \pm 1.645 \times 0.0275 = 0.66 \pm 0.045 = (0.615, 0.705)$$

Q: A company wants to estimate the average number of hours employees work per week. A sample of 36 employees shows a mean of 41 hours with a sample standard deviation of 6 hours. Construct a 99% confidence interval for the mean.

$$41 \pm 2.75 \times \frac{6}{\sqrt{36}} = 41 \pm 2.75 \times 1 = 41 \pm 2.75 = (38.25, 43.75)$$

Q: Two stores want to compare average daily sales. Store A: $$n=40$$, $$\bar{x}_A=5200$$, $$s_A=600$$; Store B: $$n=35$$, $$\bar{x}_B=4900$$, $$s_B=700. $$Construct a 95% confidence interval for the difference in means.

$$(5200 - 4900) \pm 1.99 \times \sqrt{\frac{600^2$}{40} + \frac{700^2$}{35}} = 300 \pm 1.99 \times 157.7 = 300 \pm 313.8 = (-13.8, 613.8)$$

Question 1

A retail company knows that the average amount spent per customer is $$52 with a population standard deviation of 12. If a $$random sample of 64 customers is selected, what is the probability that the sample mean spending is greater than $55? Use the Central Limit Theorem.

Accepted Answer

$$P(\bar{X} \u003e 55) = P\left(Z \u003e \frac{55 - 52}{12/\sqrt{64}}\right) = P(Z \u003e 2) = 0.0228$$

Question 2

A manufacturing company finds that 18% of its products are returned. If a random sample of 150 products is selected, what is the probability that more than 25 products are returned? (Use normal approximation to the binomial.)

Accepted Answer

$$P(X \u003e 25) = P\left(Z \u003e \frac{25.5 - 27}{\sqrt{150 	imes 0.18 	imes 0.82}}ight) = P(Z \u003e -0.41) = 0.6591$$

Question 3

The average delivery time for a courier company is normally distributed with mean 40 minutes and standard deviation of 8 minutes. If 36 deliveries are randomly selected, what is the probability that the sample mean delivery time is between 38 and 42 minutes?

Accepted Answer

$$P(38 \u003c \bar{X} \u003c 42) = P\left(\frac{38-40}{8/\sqrt{36}} \u003c Z \u003c \frac{42-40}{8/\sqrt{36}}\right) = P(-1.5 \u003c Z \u003c 1.5) = 0.8664$$

Question 4

A bank reports that 62% of customers use online banking. If a random sample of 200 customers is selected, what is the probability that the sample proportion is less than 0.58?

Accepted Answer

$$P(\hat{p} \u003c 0.58) = P\left(Z \u003c \frac{0.58 - 0.62}{\sqrt{0.62 	imes 0.38 / 200}}ight) = P(Z \u003c -1.45) = 0.0735$$

Question 5

A random sample of 49 transactions at a grocery store had a mean purchase of $$84 with a sample standard deviation of 14. $$Construct a 95% confidence interval for the mean purchase.

Accepted Answer

$$84 \pm t_{0.025,48} 	imes \frac{14}{\sqrt{49}} = 84 \pm 2.01 	imes 2 = 84 \pm 4.02 = (79.98, 88.02)$$

Question 6

A survey of 300 customers found that 198 were satisfied with the company’s service. Construct a 90% confidence interval for the proportion of satisfied customers.

Accepted Answer

$$\hat{p} = 0.66,\ SE = \sqrt{0.66 	imes 0.34 / 300} = 0.0275$$ 
$$0.66 \pm 1.645 	imes 0.0275 = 0.66 \pm 0.045 = (0.615, 0.705)$$

Question 7

A company wants to estimate the average number of hours employees work per week. A sample of 36 employees shows a mean of 41 hours with a sample standard deviation of 6 hours. Construct a 99% confidence interval for the mean.

Accepted Answer

$$41 \pm 2.75 	imes \frac{6}{\sqrt{36}} = 41 \pm 2.75 	imes 1 = 41 \pm 2.75 = (38.25, 43.75)$$

Question 8

A soft drink company claims that the average bottle contains 500 mL. A random sample of 40 bottles has a mean of 492 mL with a sample standard deviation of 20 mL. Test the claim at the 5% significance level.

Accepted Answer

$$t = \frac{492 - 500}{20/\sqrt{40}} = \frac{-8}{3.16} = -2.53$$;\ p \u003c 0.05$, so reject the claim.

Question 9

A company claims that at least 70% of customers renew their subscriptions. In a sample of 120 customers, 78 renewed. Test the claim at $$\alpha = 0.10.$$

Accepted Answer

$$H_0$$: p \geq 0.70,\ H_a: p  0.10$$,\ fail to reject H_0$.

Question 10

Two stores want to compare average daily sales. Store A: $$n=40$$, $$\bar{x}_A=5200$$, $$s_A=600$$; Store B: $$n=35$$, $$\bar{x}_B=4900$$, $$s_B=700. $$Construct a 95% confidence interval for the difference in means.

Accepted Answer

$$(5200 - 4900) \pm 1.99 	imes \sqrt{\frac{600^2$}{40} + \frac{700^2$}{35}} = 300 \pm 1.99 	imes 157.7 = 300 \pm 313.8 = (-13.8, 613.8)$$

Question 11

Two advertising campaigns were tested. Campaign Online: $$n=200$$, $$x=48$$; TV: $$n=180$$, $$x=30. $$Construct a 90% confidence interval for the difference in proportions.

Accepted Answer

$$\hat{p}_1 = 0.24,\ \hat{p}_2 = 0.167$$ 
$$SE = \sqrt{\frac{0.24 	imes 0.76}{200} + \frac{0.167 	imes 0.833}{180}} = 0.0406$$ 
$$0.24 - 0.167 \pm 1.645 	imes 0.0406 = 0.073 \pm 0.0668 = (0.006, 0.140)$$

Question 12

Two factories produce batteries. Factory A: $$n=30$$, $$\bar{x}_A=420$$, $$s_A=40$$; Factory B: $$n=35$$, $$\bar{x}_B=395$$, $$s_B=50. $$Test whether the mean lifetimes are different at $$\alpha = 0.05.$$

Accepted Answer

$$H_0$$: \mu_A = \mu_B$$ 
SE$$ = \sqrt{\frac{$$40^2$$}{30} + \frac{$$50^2$$}{35}} = 11.94$$ 
t$$ = \frac{420 - 395}{11.94} = 2.10$$ 
df$$ \approx 62$$, p = 0.039$$ \u003c 0.05$$,\ so means are significantly different.

Question 13

A company tests two website designs. Old: 250 visitors, 40 purchases; New: 300 visitors, 66 purchases. Test whether the new website has a higher purchase rate at $$\alpha = 0.05.$$

Accepted Answer

$$H_0$$: p_{new} \leq p_{old}$$ 
$$\hat{p}_{old} = 0.16$$, $$\hat{p}_{new} = 0.22$$ 
SE$$ = \sqrt{\frac{0.16 	imes 0.84}{250} + \frac{0.22 	imes 0.78}{300}} = 0.034$$ 
z$$ = \frac{0.22 - 0.16}{0.034} = 1.76$$ 
p = 0.039$$ \u003c 0.05$$,\ so the new website has a higher rate.

Question 14

Which of the following best describes the Central Limit Theorem as applied in these problems?

Accepted Answer

For large sample sizes, the sampling distribution of the sample mean (or proportion) is approximately normal, regardless of the population's distribution.

Question 15

When constructing a confidence interval for a population mean with unknown population standard deviation and sample size less than 30, which distribution should be used?

Accepted Answer

The $$t-$$distribution with $$n-1$$ degrees of freedom.

Question 16

In hypothesis testing, what does a $$p-$$value represent?

Accepted Answer

The probability of observing a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true.

Question 17

A 95% confidence interval for a mean is calculated as $$(80, 90). $$Which of the following is a correct interpretation?

Accepted Answer

We are 95% confident that the true population mean lies between 80 and 90.

Question 18

Which of the following scenarios requires using a two-proportion $$z-$$test?

Accepted Answer

Comparing the proportion of customers who purchased after two different advertising campaigns.

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