STAT 226 Supplemental InstructionPractice Exam 1

TRUE or FALSE: Circle either T or F.

  1. T/F: For a normal distribution, an observation with a z-score of -1.5 is less unusual than an observation with a z-score of 1.
  2. T/F: Categorical variables take numerical values for which arithmetic operations such as adding and averaging make sense.
  3. T/F:As the sample size increases, the population standard deviation decreases.
  4. T/F:The Center Limit Theorem can always be used if the sample size is larger than 15.
  5. T/F:The mean and standard deviation are sensitive to outliers.
  6. T/F:For any normally distributed random variable, we know that approximately 68% of the observations will be further than 1 standard deviation away from the mean.
  7. T/F:An observation larger than the mean has a positive z-score.
  8. T/F:The numbers 3, 4, 5 have the same standard deviation as 1003, 1004, 1005.
  9. T/F:In a simple random sample, each set of n individuals has an equal chance of selection.
  10. T/F:The z-score tells us how many means we are away from the standard deviation.
  11. T/F:The area under a density curve is always 1.
  12. T/F:The empirical rule only applies to a standard normal distribution.

Short Answer

Apple’s New Product. The Apple Computer Company’s sales team is interested in launching a new product in the United States. Before doing so, Apple wants to estimate the proportion of college students around the U.S. that consider busying the new product. They pick one school at random from each state and survey a simple random sample of 40 students from the selected school. Their survey results indicated that 34% of the surveyed students would buy the product at the proposed price. For this situation identify the following:

STAT 226 Supplemental InstructionPractice Exam 1

a)Population: All United States college students

b)Parameter: proportion of all U.S. college students that consider buying the new product

c)Sample: 40 students from each state

d)Statistic: 34% of the students that would buy the product at the proposed price

STAT 226 Supplemental InstructionPractice Exam 1

Use the following data set;

1, 8, 8, 8, 9, 12, 14, 14, 15, 15, 17

a)Compute the Mean

11

b)Compute the Median

12

c)Compute the 5-Number Summary

Min: 1Q1: 8Median: 12 Q3: 15Max: 17

d)Compute the Inter-Quartile Range

15-8=7

e)Name two methods that would be deemed appropriate for displaying this data. What about the data allows you to reason that these are appropriate visual displays?

*multiple answers will work as long as you can justify it properly*

f)What would you consider to be appropriate measures of center and spread for this data? Why?

Median and IQR, they are not sensitive to the outlier of 1.

For this question, you must use the Empirical Rule. After finishing school, you decide to open a sunglass-selling stand. Your stand is only open during the summer. You know that based on other sunglass selling stand in the area, that the overall summer sales of sunglasses for this area is approximately Normally distributed with a mean of 500 sunglasses and a standard deviation of 65 sunglasses. Assuming your stands sales will follow the same distribution, answer the following;

a)What is the probability that you sell more than 695 sunglasses?

695=500+65x  x=3  3 standard deviations above the mean .0015 or .15%

b)Your goal for the first summer is to sell as many glasses as the middle 95% of all such stands. Specify the range in which the number of glasses that you sell over the summer has to fall inside.

Middle 95% means within 2 std dev. Away from the mean  500+(2*65)=370 

500-(2*65)=630370 &630

c)You sell sunglasses for $15 each and would like your total sales for the summer to be at least $6,525. What is the probability that you achieve this goal?

6525/15=435435=500+65xx= -11-16%=84%

d)What price must you charge to sell enough sunglasses to fall at the 2.5th percentile but still make at least $9,250?

2.5% is 2 std dev. Below mean500-(2*65)=3709250/370=$25/pair

IQ Test. You and your friends decide that you want to know your IQ scores. You want to compare your IQ scores to the national averages. The mean of normally distributed distribution of IQ’s in the United States is 100 and the standard deviation is 13. Use this information to answer the following questions.

a)What percent of people score under 120?

(120-100)/13=1.54(z-score)find .9382 in table A93.82%

b)What percent individuals score between 90 and 115?

(90-100)/13=.77(z-score)find .2206 in table A(115-100)/13=1.15(z-score)find .8749 in table A.8749-.2206=65.43%

c)Your friend Max brags that he is smarter than 90% of the population. He scores a 118, did he accomplish this?

(118-100)/13=1.38(z-score)find .9162 in table A91.62%>90% yes

d)Your friend is a National Merit Scholar and scores a 135, what percentage of people score higher?

(135-100)/13=2.69(z-score)find .9964 in table A1-.9964=.0036.36%

e)Which is more unusual, a score of 114 or 87?

114 (larger distance from mean)

New Phone. The Manufacturer of a new cell phone had “the world’s fastest-charging battery by far.” Extensive laboratory testing has shown that the distribution for the time it takes the new phone to chargecompletely has a mean of 50minutes and a standard deviation of 3.5 minutes. Suppose 50 of these cameras are randomly selected from the production line and tested. The time until the image is revealed is recorded for each.

a)Find the probability that we will obtain a sample mean greater than 49.2 minutes.

(49.2-50)/(3.5/(501/2))= -1.62(z-score)find .0562 in table A1-.0562=94.74%

b)What is the mean time until the phone is charged for 50 sampled phones such that only 3% of every mean of sample size 50 will be faster?

Reverse lookup .0300 in table A to find -1.88(z-score) -1.88=(x-50)/(3.5/(501/2))x=49.07 minutes

c)What percentage of phones fall between 48.9 and 50.3 minutes for charging time.

(48.9-50)/(3.5/(501/2))= -2.22(z-score)find .0136 in table A(50.3 -50)/(3.5/(501/2))= .61(z-score)find .7191 in table A.7191-.0136=71.55%

d)Refer to part (c). Describe the changes in the sampling distribution of the sample mean if the sample size is decreased from n=50 to n=20.

Range is wider with a smaller sample size, less precision