Q: A school district tests 133 kindergarteners for nearsightedness, which affects 12% of children. What is the standard deviation of the sampling distribution of the sample proportion?

$$\sqrt{\frac{0.12 \times 0.88}{133}}$$

Q: In a sample of 250 U.S. adults, what is the mean and standard deviation of the sampling distribution for the sample proportion of obesity, assuming the true rate is 36%?

Mean: $$0.36$$, SD: $$\sqrt{\frac{0.36 \times 0.64}{250}}$$

Q: A company wants to estimate the proportion of 16- to 24-year-old cell phone owners who go online, to within $$\pm 7.5\%$$ with 95% confidence. What is the minimum sample size required (using $$p = 0.5$$ for maximum variability)?

$$n = \left(\frac{1.96}{0.075}\$$right)^2$$ \times 0.5 \times 0.5$$

Q: A textbook editor wants to estimate the mean word length to within $$0.5$$ letters with 98% confidence. If the sample standard deviation is $$2.7$$, what is the minimum sample size required?

$$n = \left(\frac{2.33 \times 2.7}{0.5}\$$right)^2$$

Question 1

A company's water treatment process produces batches with a mean chemical concentration of 75 ppm and a standard deviation of 4.2 ppm, modeled as Normal. What percent of batches exceed the legal limit of 80 ppm?

Accepted Answer

About 6.87%

Question 2

A manufacturer wants no more than 2% of water batches to exceed 80 ppm of a pollutant. If the standard deviation remains at 4.2 ppm, to what mean should the machine be set? (Use the standard Normal distribution.)

Accepted Answer

$$71.3$$

Question 3

The lifespans of a brand of graphing calculator are normally distributed with mean $$620$$ days and standard deviation $$82$$ days. If the company sets the warranty at $$540$$ days, what proportion of calculators will they have to replace?

Accepted Answer

About 10.5%

Question 4

A school district tests 133 kindergarteners for nearsightedness, which affects 12% of children. What is the standard deviation of the sampling distribution of the sample proportion?

Accepted Answer

$$\sqrt{\frac{0.12 	imes 0.88}{133}}$$

Question 5

In a sample of 250 U.S. adults, what is the mean and standard deviation of the sampling distribution for the sample proportion of obesity, assuming the true rate is 36%?

Accepted Answer

Mean: $$0.36$$, SD: $$\sqrt{\frac{0.36 	imes 0.64}{250}}$$

Question 6

Researchers sample 50 frogs from a population with mean weight $$118g $$and standard deviation $$14g. $$Their sample mean is $$110g. $$What is the $$z-$$score for this sample mean?

Accepted Answer

$$z = \frac{110 - 118}{14/\sqrt{50}}$$

Question 7

A random sample of 450 university applications finds 46 private school students. Construct a 90% confidence interval for the proportion of private school applicants.

Accepted Answer

$$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$ where $$\hat{p} = \frac{46}{450}$$ and $$z^* = 1.645$$

Question 8

A company wants to estimate the proportion of 16- to 24-year-old cell phone owners who go online, to within $$\pm 7.5\%$$ with 95% confidence. What is the minimum sample size required (using $$p = 0.5$$ for maximum variability)?

Accepted Answer

$$n = \left(\frac{1.96}{0.075}\$$right)^2$$ 	imes 0.5 	imes 0.5$$

Question 9

A hand-held sensor is tested for detecting explosives. Out of 50 trials (randomly placing the explosive in one of 4 boxes), the technician finds the explosive 16 times. What is the null hypothesis for a significance test of the device's effectiveness?

Accepted Answer

$$H_0$$: p = 0.25$$

Question 10

A political party samples 853 Oregon adults and finds 113 are Hispanic or Latino. The 2010 Census reported 11.7%. What is the appropriate alternative hypothesis to test if the proportion has changed?

Accepted Answer

$$H_a: p 
eq 0.117$$

Question 11

A textbook editor wants to estimate the mean word length to within $$0.5$$ letters with 98% confidence. If the sample standard deviation is $$2.7$$, what is the minimum sample size required?

Accepted Answer

$$n = \left(\frac{2.33 	imes 2.7}{0.5}\$$right)^2$$

Question 12

A student samples 30 people and finds a mean of 4.5 movies seen per year, with a standard deviation of 1.2. The national average is 3.9. What is the $$t-$$statistic for testing if this sample provides evidence of higher attendance?

Accepted Answer

$$t = \frac{4.5 - 3.9}{1.2/\sqrt{30}}$$

Question 13

Which of the following best describes what a 95% confidence interval means in the context of estimating a population proportion?

Accepted Answer

If we repeated the sampling process many times, about 95% of the calculated intervals would contain the true population proportion.

Question 14

A company wants to reduce the standard deviation of a process to decrease the proportion of out-of-spec batches. In this context, what does achieving a smaller standard deviation mean?

Accepted Answer

The process produces more consistent results, with less variation from batch to batch.

Question 15

In hypothesis testing, what does the P-value represent in the context of testing a new sensor for explosives?

Accepted Answer

The probability of observing a result as extreme as, or more extreme than, the one obtained, assuming the sensor is no better than random guessing.

Question 16

A sample of 20 words from a statistics textbook has a mean word length of 5.6 letters. The editor wants to test if the mean is less than the target of 6.5 letters. What is the correct null hypothesis?

Accepted Answer

$$H_0$$: \mu = 6.5$$

Question 17

A 95% confidence interval for the mean number of movies seen per year is calculated as $$[4.06, 4.94]. $$What is the correct interpretation?

Accepted Answer

We are 95% confident that the true mean number of movies seen per year by this population is between 4.06 and 4.94.

Unit 3 Review: Step-by-Step Guidance for Statistics Questions

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