1. Find the area under the standard normal curve (round to four decimal places)
a. To the left of z=1.65
b. To the right of z = 0.54
c. Between z = -2.05 and z = 1.05
2. Find the z-score that has an area of 0.23 to its right.
3. The average height of a certain group of children is 49 inches with a standard deviation of 3 inches. If the heights are normally distributed, find the probability that a randomly selected child’s height will be: (note: round to four decimal places)
a. Less than 45 inches.
b. Between 50 and 55 inches.
4. Scores on a statistics were normally distributed with a mean of 72 and a standard deviation of 9. Find the 80th percentile of the scores.
5. A population has mean m = 184 and s = 33 . Find m x and s x for samples of size n = 64 . (round s x to three decimal places).
6. A sample of size 25 will be drawn from a skewed population with mean 60 and standard deviation 7. Is it appropriate to use the normal distribution to find probabilities for x ? Explain your answer.
7. Among all the state income tax forms filed in a particular state, the mean income tax paid was m = $1950 and the standard deviation was s = $490 . As part of a study of the impact of a new tax law, a sample of 100 income tax returns is examined. What is the probability that the sample mean of these 100 returns is greater than $2050? Would this be unusual? Round your probability to four decimal places.
8. A sample of size n = 137 is drawn from a population with a proportion p = 0.16 . Let ˆp be the sample proportion.
a. What specific conditions must be met in order to use the standard normal distribution to find probabilities for ˆp?
b. Find m ˆp and s ˆp . Round s ˆp to four decimal places.
c. Find P(ˆp < .13). Round to four decimal places.
9. The amount of time that a certain cell phone will keep a charge is known to be normally distributed with a standard deviation s = 15 hours. A sample of 45 cell phones has a mean time of 145 hours. Let m represent the population mean time that
a cell phone will keep a charge.
a. What is the point estimate of m ?
b. What is the standard error of the point estimate? Round to three decimal places.
c. Suppose that a 90% confidence interval is to be constructed for the mean time. Find the margin of error for this confidence interval (use three decimal places).
e. Construct the 90% confidence interval and interpret your result:
f. What sample size would be necessary so that a 98% confidence interval will have a margin of error of 2 hours?
10. Circle the correct word: If we decrease the sample size and keep the confidence level the same, we increase / decrease the margin of error.
11. Use Table A.3 to find the critical value ta /2 needed to construct a confidence interval of the given level with the given sample size:
a. Level 99%, sample size 19
b. Level 90%, sample size 172
12. In a sample of 133 hip surgeries of a certain type, the average surgery time was 130.2 minutes with a standard deviation of 24.6 minutes. Construct a 90% confidence interval for the mean surgery time for this procedure and interpret your result. Round your answer to one decimal place.
13. Scores on a standardized reading test in a certain school district have a mean of 75. After modifications to the reading curriculum were made, an educator believes that the mean may now higher. State the appropriate null and alternate hypotheses.
14. It is desired to check the calibration of a scale by weighing a standard 10-gram weight 100 times. Let m be the population mean reading on the scale, so that the scale is in calibration if m=10 and out of calibration if m¹10. A test is made of the hypotheses H 0 : m = 10 versus H 1 : m ¹ 10 . Consider three possible conclusions: (i) The scale is in calibration. (ii) The scale is not in calibration. (iii) The scale might be in calibration.
a) Which of these three conclusions is best if H0 is rejected?
b) Assume that the scale is in calibration, but the conclusion is reached that the scale is not in calibration. Which type of error is this?
15. A test is made of H0 :m=40 versus H1:m>40. A sample of size n=75 is drawn, and x = 45 . The population standard deviation is s = 16 .
a. Compute the value of the test statistic z .
b. Is H 0 rejected at the a = 0.05 level?
16. For the following MINITAB output:
a. What are the null and alternate hypotheses?
b. What is the P-value?
c. Using the P-value approach, do you reject H 0 at the a = 0.05 level? State a conclusion.
17. For the past several years, the mean number of people in a household has been declining. A social scientist believes that in a certain large city, the mean number of people per household is less than 2.5. To investigate this, she takes a simple random sample of 250 households in the city, and finds that the sample mean number of people is 2.3 with a sample standard deviation of 1.3. Can you conclude that the mean number of people per household is less than 2.5?
a. State the null and alternate hypotheses.
b. Should we perform a z-test or a t-test? c. Compute the value of the test statistic.
d. Do you reject H0 at the a = 0.01 level?
e. State a conclusion.
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