Question

Researchers investigating characteristics of gifted children collected data from schools in a large city on a random sample of thirty-six children who were identified as gifted children soon after they reached the age of four. The following histogram shows the distribution of the ages (in months) at which these children first counted to 10 successfully. Also provided are some sample statistics.
USE THE TI CALCULATOR (Z-TEST and Z-INTERVAL) for the needed values.

n	36
min	21
mean	30.69
sd	4.31
max	39

(a) Suppose you read online that children first count to 10 successfully when they are 32 months old, on average. Perform a hypothesis test to evaluate if these data provide convincing evidence that the average age at which gifted children first count to 10 successfully is less than the general average of 32 months. Use a significance level of α = 0.10.


The correct hypotheses for the test are:

• Ho: μ = 10
  Ha: μ < 10
• Ho: μ = 32
  Ha: μ ≠ 32
• Ho: μ = 32
  Ha: μ < 32
• Ho: μ = 10
  Ha: μ ≠ 10

The test statistic is z =  (please round to two decimal places)


The p-value for the test is p =  (please round to four decimal places)


The result of the hypothesis test is:

• Since p ≥ α we accept the null hypothesis
• Since p < α we accept the null hypothesis
• Since p ≥ α we do not have enough evidence to reject the null hypothesis
• Since p < α we reject the null hypothesis and accept the alternative

(c) We are 90% confidence that the average age at which gifted children first count to 10 successfully is between  months and  months. (please round to one decimal place)



(e) Do your results from the hypothesis test and the confidence interval agree?

• no
• yes

Answer 1

Solution :

Given that, sample size = 36, sample mean = 30.69

standard deviation = 4.31

The null and alternative hypotheses are,

Ho : μ = 32

Ha : μ < 32

Third option is correct

The test statistic is,

The test statistic is, z = -1.82

p-value = P(Z < -1.82) = 0.0344

The p-value for the test is p = 0.0344

The result of the hypothesis test is :

Since p < a we reject the null hypothesis and accept the alternative.

Last option is correct

c) A 90% confidence level has significance level of 0.10 and critical value is, z^* = 1.645

The 90% confidence interval for population mean is,

​​​

​​​​​​Therefore, we are 90% confidence that the average age at which gifted children first count to 10 successfully is between 29.5 months and 31.9months.

e) Yes, we agreed to the results from the hypothesis test and the confidence interval.

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